WATCH: DHS Secretary Mayorkas delivers remarks at U.S.-Mexico border

Department of Homeland Security Secretary Alejandro Mayorkas held a news briefing on Monday at the U.S.-Mexico border to discuss immigration as the Senate continues to negotiate a proposal linking funding for the border to Ukraine aid.

Watch Mayorkas’ remarks in the player above.

Senate negotiators were trying to land a bipartisan border security proposal this week that could unlock Senate Republican support for Ukraine aid. But as Congress returns, House conservatives are trying to interject their own hardline immigration demands.

Senate negotiators met Monday morning as they raced to finish work on legislative text. They were hoping this week to present the details of a bipartisan bill aimed at reducing the number of migrants who travel to the southern border to apply for asylum protections in the U.S. The small group of senators has been working for months on the legislation, but disagreements remained.

“We’re certainly narrowing down the issues,” Sen. Chris Murphy, who has been leading the Democratic side of the negotiation, told reporters. “We’re hopeful we’ll have something to present to our colleagues soon.”

President Joe Biden’s administration has also been directly involved in the talks as the president tries to both secure support for a top foreign policy priority — funding Ukraine’s defense against Russia — and demonstrate action on a potential political weakness — his handling of the historic number of migrants seeking asylum at the U.S. border with Mexico.

Biden has faced staunch resistance from conservatives to his $110 billion request for a package of wartime aid for Ukraine and Israel as well as other national security priorities. In the Senate, Republicans have demanded that the funding be paired with border security changes.

But in the House, conservatives have criticized any proposals that fall short of the strict border measures they passed on a party-line vote last year. And some House members aligned with Donald Trump, the former president and current Republican presidential front-runner, have suggested they would not support any bipartisan proposal — no matter the substance — if it means giving Biden a border bill to sign in an election year.

House Speaker Mike Johnson, joined by about 60 fellow House Republicans, visited a Texas border city last week to press the case that the border legislation should tack closely to what the House has already passed. Johnson said the visit only increased his resolve to ensure border policy changes are included with Ukraine funding and called the House bill a “necessary ingredient.”