WATCH: Senate convenes as Trump ramps up pressure to kill the filibuster and end the shutdown

WASHINGTON (AP) — With talks to reopen the government intensifying, Senate Majority Leader John Thune on Thursday expressed determination to move ahead with a new bipartisan package of funding bills, a renewed effort to win over enough Democrats and end the shutdown that is now in its 37th day.

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It was unclear if the move could force a resolution. Senate Democrats, who have now voted 14 times not to reopen the government, left their second caucus meeting of the week with few answers about how to proceed and whether they could find a compromise with Republicans — or even with each other — on how to end the shutdown.

Some Democrats say the fight isn’t over until Republicans and President Donald Trump agree to extend health care subsidies that expire in January. Others are pushing a deal that would reopen the government with only an agreement for a future vote on the issue.

READ MORE: 4 tips for navigating higher ACA health care premiums

“Working on unity and working on health care,” Democratic Sen. Sheldon Whitehouse of Rhode Island said after the meeting.

Still, lawmakers in both parties were feeling increased urgency to alleviate the growing crisis at airports, pay government workers and restore delayed food aid to millions of people now that the shutdown has become the longest in U.S. history.

Thune’s decision to keep the Senate in session, perhaps over the weekend, came after President Donald Trump urged Senate Republicans at a White House breakfast on Wednesday to end the shutdown. Trump said he thought the six-week impasse was a “big factor, negative” in Tuesday’s elections that were overwhelmingly favorable for Democrats.

A new effort to reopen the government

The bipartisan package Thune is proposing would fund parts of government — food aid, veterans programs and the legislative branch, among other things — and extend funding for everything else until December or January. Procedural votes on the proposed plan could begin as soon as Friday and serve as a test of Democratic support.

The new package would replace the House-passed bill that the Democrats have repeatedly rejected. That legislation would only extend government funding until Nov. 21, a date that is rapidly approaching after six weeks of inaction.

The details were still to be worked out, but the new legislation mirrors a tentative plan that the moderate Democrats have been sketching out in hopes of finding agreement. The proposal led by New Hampshire Sen. Jeanne Shaheen would also take up Republicans on their offer to hold a vote on extending the expiring Affordable Care Act subsidies at a later date.

It was still unclear what Thune, who has refused to negotiate while the government is closed, would promise on health care and if enough Democrats would agree to move ahead. Republicans have for weeks been five votes short of the 60 they need.

Democrats divided after election wins

Democrats are facing pressure from both unions eager for the shutdown to end and from allied groups that want them to hold firm. Many Democrats have argued that strong results for Democrats in Tuesday’s election show voters want them to continue the fight until Republicans yield and agree to extend the health tax credits.

A vote on the health care subsidies “has got to mean something,” Vermont Sen. Bernie Sanders, an independent who caucuses with the Democrats, said this week. “That means a commitment by the speaker of the House, that he will support the legislation, that the president will sign.”

But Speaker Mike Johnson, R-La., made clear Thursday morning he won’t make any commitment to Democrats. “I’m not promising anybody anything,” Johnson said when asked if he could promise a vote on a health care bill.

WATCH: Johnson says he won’t promise ACA vote in the House as part of a shutdown deal

Johnson’s clear refusal was a setback for negotiators. Michigan Sen. Gary Peters, one of the moderate Democrats involved in negotiations, said his comments were “a significant problem.”

“We have to make sure we have a deal that we can get broad support for,” Peters said.

Senate Democratic leader Schumer has not yet weighed in on the latest push. He has repeatedly called for Trump to sit down with Democrats — a meeting that seems unlikely to happen.

“Donald Trump clearly is feeling pressure to bring this shutdown to an end,” Schumer said Thursday.

Trump has also been increasingly fixated on pushing Republicans to scrap the Senate filibuster to speed reopening — a step many GOP senators, including Thune, reject out of hand. Trump kept up the pressure in a video Wednesday, saying the Senate’s 60-vote threshold to pass legislation should be “terminated.”

Closed-door negotiations become public

Democrats and Republicans insisted they were making steady progress on a deal.

In a new development on Thursday, Republicans suggested that they might be open to including language in a final agreement that would reverse some mass firings of government workers by the White House, according to two people familiar with the private talks granted anonymity to discuss them. But it was unclear if that proposal would be included in the new package of bills.

Senate Appropriations Committee Chairwoman Susan Collins, a moderate Republican who has been talking to Democrats, says she wants furloughed workers to be given back pay and for workers who have been fired during the shutdown to be “recalled.”

“We’re still negotiating that language,” she said.

Associated Press writer Lisa Mascaro contributed to this report.