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Congressional leaders left a meeting at the White House on Wednesday signaling cautious but fresh hope that a bipartisan deal could be reached to fund Ukraine and overhaul policies at the southern border.

Both Senate Majority Leader Chuck Schumer, D-N.Y., and Minority Leader Mitch McConnell, R-Ky., told reporters after the meeting that a deal on supplemental security funding could come for a vote soon.

"I am more optimistic now that we can come to an agreement on border and Ukraine in one package, along with aid to Israel, along with humanitarian aid for the Palestinians in Gaza, and along with helping Indochina," Schumer said. "I put the chances a little bit greater than half now. And that's the first time I can say that."

McConnell called it a "constructive discussion" and added a deal could "be on the floor next week."

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Johnson, Schumer

Speaker Mike Johnson and Senate Majority Leader Chuck Schumer were two of the congressional leaders meeting President Biden at the White House on Wednesday.

"We've been talking about this for a very long time. It’s time to try to act," he said upon returning to the U.S. Capitol on Wednesday.

President Biden expressed similar optimism on Thursday.

Asked by Fox News' Jacqui Heinrich whether there were any "sticking points" left in coming to an agreement, Biden said, "I don't think we have any sticking points left."

Wednesday's meeting was intended to be focused on Democrats' $110 billion supplemental aid request for Ukraine, Israel and others. But Republicans have insisted on border and immigration policy reforms for their support, as the border crisis becomes an increasingly bipartisan issue.

But while the Senate has been discussing a bipartisan path forward on border policy, the House GOP majority has been steadfast in calling for nothing short of the provisions in H.R.2, which passed the House last year and includes Trump administration-era border policies like Remain in Mexico and construction of a border wall.

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Democrats who control the Senate and White House have called the bill a nonstarter.

A top House Republican who emerged from the meeting, Foreign Affairs Committee Chairman Mike McCaul, R-Texas, was cautiously encouraged that President Biden was receptive to border policy changes – and suggested his conference could be flexible as well.

Mike Johnson

Johnson was joined by, from left, House Intelligence Committee Chair Mike Turner, Armed Services Committee Chair Mike Rogers, and Foreign Affairs Committee Chair Mike McCaul. (Getty Images)

"He said that I am ready to make significant changes to the border. He said it's broken, he knows that, and it needs to be fixed," McCaul said of Biden. 

He said he and Speaker Mike Johnson, R-La., specifically pushed for the Remain In Mexico provision.

"I think that would be a significant policy change, that would get to the heart of the problem. That is what drives the cartels," McCaul said. "You stop that, you stop the flow and you solve the problem."

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McCaul said of the current status of talks, "Of course, we're making the pitch for all of H.R.2. I also live in a realistic world."

Johnson called the meeting "productive" upon leaving the White House.

Later on the "Ingraham Angle," he signaled Republicans were still pushing for H.R.2 provisions but said of Biden at the meeting, "He said we're ready to do big things on the border."

Texas National Guard at border

A National Guard soldier on the banks of the Rio Grande at Shelby Park on Jan. 12, 2024, in Eagle Pass, Texas. (Brandon Bell/Getty Images)

Rep. Jim Himes, D-Conn., the top Democrat on the House Intelligence Committee, told reporters that Johnson never specifically said it was H.R.2 or nothing during the meeting.

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He did say Johnson insisted "the most important issue for the American people is the border" and that he defended specific measures of the bill. 

The White House said of the meeting, "The President also made clear that we must act now to address the challenges at the border. He said he is encouraged by the progress being made in the bipartisan negotiations happening in the Senate. He expressed his commitment to reaching a bipartisan agreement on border policy and the need for additional resources at the border. The President called on Congress to swiftly pass his full national security supplemental."

Fox News' Brianna O'Neil contributed to this report