WILLIAM BRANGHAM: It is a delicate and momentous moment in the Middle East.

The Israeli military tonight says it has launched a new operation into parts of Rafah in Southern Gaza.

At the same time, the Israeli government says it's continuing negotiations to reach a cease-fire in its war on Hamas in exchange for the release of Israeli hostages.

It has been a day with fast-moving, competing headlines.

And, thankfully, Nick Schifrin is here to break it all down for us.

So, Nick, what is the latest on the hostage negotiations?

NICK SCHIFRIN: The headline tonight, William, is that negotiations continue.

But what got us here is not a straight line.

Late last week, U.S., Egyptian and Qatari mediators offered Hamas what the U.S. called a -- quote - - "extraordinarily generous proposal approved by Israel."

That would be a period of calm for 40 days.

Hamas would release 33 hostages, the female, the elderly and the infirm.

Israel would lease some 700 Palestinian detainees.

And after those releases, Gazans would have freedom of movement within Gaza.

And the two sides would discuss a permanent cease-fire and withdrawal of troops from Gaza.

Now, early today Hamas released a statement of its -- quote -- "approval" of an Egyptian and Qatari proposal regarding the cease-fire agreement.

Israeli officials quickly told me that Hamas had not approved anything or what had it approved was not the deal on the table.

The reality is somewhere in the middle, William.

U.S. officials confirmed that Hamas responded to the proposal, didn't accept anything.

At the same time, their response is being called positive by regional officials that I speak to.

We do not have the exact details of how Hamas is asking for the proposal to be changed.

What we do know is that the Israeli war cabinet has released a statement about what Hamas said today.

And that statement is -- quote -- "Although the Hamas proposal is far from Israel's position, Israel will send a delegation of working-level mediators to exhaust the possibility of reaching an agreement under conditions acceptable to Israel."

So, Israel has neither accepted nor denied Hamas' proposal today.

And so the negotiations continue.

WILLIAM BRANGHAM: So, meanwhile, Israel has always been talking about going into Rafah.

What is the latest on that?

NICK SCHIFRIN: The war cabinet in that statement also said that it would continue military operations in Rafah to -- quote - - "exert military pressure on Hamas."

And there are early reports right now that, in fact, Israeli ground troops, including tanks, have crossed from Israel toward Rafah.

Israeli officials tell me this is a -- quote -- "targeted operation" designed to keep the negotiations pressure on Hamas.

This is not the Rafah operation that you and I and U.S. officials and Israeli officials have been talking about, that Israel says is required to kill the last four battalions of Hamas and its about a few thousand fighters that are part of that.

And the U.S. continues to reject that idea for a Rafah operation, as John Kirby, national security spokesman, said today.

JOHN KIRBY, NSC Coordinator For Strategic Communications: We have been very clear that we don't support a major ground operation in Rafah, operations in general that put at greater risk the more than a million people that are sheltering there.

NICK SCHIFRIN: Asked if Israel had presented the administration a -- quote -- "comprehensive plan" to evacuate so many people from Rafah and assault the city, Kirby had a one-word answer: "No."

WILLIAM BRANGHAM: Meanwhile, the fate of this hostage deal seems intertwined with what could or could not unfold in Rafah.

Why is this so a delicate moment?

NICK SCHIFRIN: On the diplomatic front, it is critical, because U.S. officials say a cease-fire and, therefore, no Rafah operation would unlock some of the regional goals that they have, including Saudi normalization.

And, today, Saudi Arabia released a pointed statement saying that any Rafah operation would - - quote -- "limit international peace efforts" and the statement actually mentioned the word genocide.

A Rafah operation could also lead the U.S. to impose restrictions on weapons to Israel.

Now, on the humanitarian front, U.S. officials say any operation in Rafah would make it impossible to deliver enough aid to more than a million people who are taking shelter of there, many of whom have been displaced multiple times.

This morning they evacuated with everything they could carry, after the Israeli army dropped leaflets from the sky urging 100,000 Gazans in Eastern Rafah to flee to Al-Mawasi, already a tent city.

MOSES KONDOWE, Project HOPE: If the offensive is to happen, it's going to be catastrophic.

NICK SCHIFRIN: Moses Kondowe is the Gaza team lead of international humanitarian organization Project HOPE.

He spoke to us last week from Rafah.

Of 1.2 million displaced in the city, half are kids.

And one in five children are malnourished.

MOSES KONDOWE: I see people who are in fear.

What I see, especially in the morning, are long, winding queues where people will be jostling to get the limited available water.

NICK SCHIFRIN: The U.N. says, in Northern Gaza, famine is imminent.

But, this weekend, World Food Program executive director Cindy McCain went further.

CINDY MCCAIN, Executive Director, World Food Program: There is famine, full-blown famine, in the north, and it's moving its way south.

NICK SCHIFRIN: Today, Prime Minister Benjamin Netanyahu agreed to a request from President Biden to open Kerem Shalom, a key crossing in Southern Israel which had been temporarily closed this weekend, William, because Hamas attacked the crossing, killing four Israeli soldiers and disrupting aid into Gaza.

But that will reopen, U.S. officials say, basically tonight or tomorrow.

WILLIAM BRANGHAM: Nick Schifrin, thank you so much.

NICK SCHIFRIN: Thank you.

WILLIAM BRANGHAM: We turn now to Aaron David Miller.

He's a senior fellow at the Carnegie Endowment for International Peace and someone who has played a key role in U.S. Middle East policy for multiple American presidents.

Aaron David Miller, great to have you back on the "NewsHour."

What do you make of what transpired today?

You heard what Nick's been reporting, Hamas saying, yes, we agreed to this deal, Israel saying, I'm not so sure what you have agreed to.

How do you see this?

AARON DAVID MILLER, Carnegie Endowment for International Peace: We went from the top of the mountain briefly to the valley below in about three hours.

My take is very simple.

Neither side is in a hurry to reach an agreement, but both want to blame the other for obstructing one.

And I think that's what you're seeing play out right now.

Hamas' proposal, the one they agreed to, is one, at least The Wall Street Journal is reporting terms that no Israeli government could accept.

And the reality is, the prime minister of Israel is under no pressure.

In fact, his coalition, frankly, would be more stable if he didn't risk Hamas for -- prisoner exchange.

And he's juggling balls, as he usually does.

The Rafah operation, or some limited operation, may or may not go forward in the days and weeks ahead.

He's also managing to send a delegation in good faith to Doha or Cairo to continue negotiations on a proposal that he's simply not prepared to accept.

And Hamas will continue to play this game as well.

They do not want to trade hostages, any hostages, for anything other than a comprehensive deal, which leads to a permanent fire, the withdrawal of Israeli forces from Gaza, and, I suspect Yahya Sinwar, ensconced in some tunnel meters below ground in Rafah, Khan Yunis, or maybe even in Sinai, surrounded by hostages, plans to survive this and to lead Hamas in the post-conflict period and play a major role in what transpires in Gaza.

So to reach a deal, parties have to be in a hurry.

And the only party that -- understandably, to be sure, that's in a hurry is the Biden administration.

WILLIAM BRANGHAM: Yes, I mean, the conditions that you're describing are things that no Israeli president, let alone Bibi Netanyahu, could say yes to.

AARON DAVID MILLER: I think that's right.

And I guess the real question is, is there any pressure that can be exerted on either party to bring them around to some middle position, which tries to square the circle of conflicting demands and political necessities that both Hamas and the government of Israel, the Netanyahu government, and the war cabinet, are operating under?

And, right now, seven months -- this is the seventh month of the war this week.

I don't see any lever, frankly, right now that can be pulled that could free hostages, de-escalate Israeli military activity, and surge humanitarian assistance into Gaza.

That's the real tragedy here.

I see no way out of this right now.

It's what I would describe to you as a strategic cul-de-sac with no way out right now.

WILLIAM BRANGHAM: I mean, given everything that you're describing there, we know the Biden administration has been trying, somewhat fruitlessly, to pressure the Israelis on this front.

What would you be counseling them to do?

What can they do to move that needle in any way?

AARON DAVID MILLER: You know, all politics is local.

And Tip O'Neill was right.

And the reality is -- I have to keep reminding myself of this every single day since October 7 -- since October 7, the trajectory of this conflict, the escalation, the de-escalation, the ambitions, the motives, how it ultimately is going to end, has basically been dominated by the two noncombatants.

What this conflict has shown is the limits of great power's capacities, even if they have the influence on paper, as the U.S. does, to alter the trajectories of two parties who believe they are in an existential conflict.

And, right now, the secretary of state is facing a very important decision this week.

He's got to certify that the assurances the Israelis have provided under the National Security Memorandum 25, that they are abiding by international humanitarian law and not obstructing humanitarian assistance into Gaza, that those assurances that the Israelis have provided are -- quote -- "credible and reliable."

There are a lot of people who don't believe that, including many in the Democratic Party, including mainstream Democratic senators.

What the secretary of state is going to decide, unclear, but, certainly, that's a potential lever.

But even if he pulled that, the question is whether or not that's going to change the trajectory of the Netanyahu government.

I don't think so.

WILLIAM BRANGHAM: All right, Aaron David Miller of the Carnegie Endowment, thank you so much for being here.

AARON DAVID MILLER: Thanks for having me.