Zelenskyy in Washington to make the case for Ukraine aid stuck in congressional limbo

Ukrainian President Volodymyr Zelenskyy visited Washington Tuesday to meet with President Joe Biden and congressional leaders, a high-stakes visit that the White House said is coming at a "critical time" with additional Ukraine aid being debated on Capitol Hill.

Zelenskyy arrived at the U.S. Capitol just before 9 a.m. for his meeting with senators flanked by Majority Leader Chuck Schumer and Minority Leader Mitch McConnell.

Exiting the room more than an hour later, Schumer called it a "productive meeting" and told reporters that Zelenskyy outlined the help he needs and how it will help Ukraine defeat Russian invaders.

Schumer also said Zelenskyy made clear he "needs the aid quickly," and said any lapse would cause European allies to question the U.S.

"He made it clear, and we all made it clear, that if we lose, [Russian President Vladimir] Putin wins and this will be very, very dangerous for the United States," Schumer said.

Zelenskyy also met with House Speaker Mike Johnson, who will be a tougher sell on getting Ukraine aid across the finish line.

Johnson called it a "good meeting" and said he reiterated to Zelenskyy that they "stand with him and against Putin's brutal invasion."

But then he took aim at the White House, saying officials haven't provided "clarity and detail" on the path forward for Ukraine to defeat Russia.

"What the Biden administration seems to be asking for is billions of additional dollars with no appropriate oversight, no clear strategy to win, and none of the answers that I think the American people are owed," Johnson told reporters. "I've also made very clear from day one that our first condition on any national security supplemental spending package is about our own national security. The border is an absolute catastrophe."

Johnson said it's "not the House's issue right now."

"The issue is with the White House and the Senate, and I implore them to do their job because the time is urgent," Johnson said. "And we do want to do the right thing here."

Zelenskyy did not respond to questions as he met with various leaders at the Capitol.

Later Tuesday, the Ukrainian president and Biden will meet at the White House and hold a joint press conference.

"We're looking forward to this visit," John Kirby, a spokesman for the White House National Security Council, told reporters on Monday. "It's a chance for the president to get an update from President Zelenskyy about how things are going on the battlefront but also to make it very clear to President Zelenskyy and the Ukrainian people that we're gonna continue to support them, particularly at this very difficult time."

Billions of dollars requested by the Biden administration to help Ukraine fight Russia's invasion, as well as to support Israel and Taiwan, are currently in limbo in a battle over immigration policy and the southern border.

Republicans in the Senate are seeking significant changes to asylum and other immigration procedures in exchange for passing the roughly $110 billion supplemental aid package.

Some GOP lawmakers have also become increasingly skeptical of the U.S. continuing to fund Ukraine, despite arguments that it helps weaken Russia, because they think that money could be better spent on domestic issues.

The clock is winding down for both sides to come to a compromise before the holiday recess begins at the end of the week.

"We are running out of resources already in the bank to continue to assist them," Secretary of State Antony Blinken said Sunday on ABC's "This Week."

McConnell, the Senate's top Republican and a staunch supporter of Ukraine, on Monday said border provisions remained top of mind for him and his colleagues.

"When it comes to keeping America safe, border security is not a side show -- it is ground zero," he said. "Senate Republicans have no more spare time to explain this basic reality. We cannot convince anyone who doesn't want to acknowledge the glaring facts on the ground. The Senate has to act."

Schumer, the Senate's top Democrat, contended in his own remarks that "Republicans and only Republicans are holding everything up because of unrealistic maximalist demands on the border. I want to be very clear: Democrats very much want an agreement if possible."

Zelenskyy arrived in Washington on Monday to give a speech at the National Defense University. There, he was introduced by Defense Secretary Lloyd Austin, who described America's security commitment to Ukraine as "unshakeable."

Zelenskyy, speaking to students, argued that delays in approving Ukraine aid plays into Putin's hands and those of his "sick clique."

"America and all free nations need to be confident in themselves, in their strengths, in their leadership so that dictatorships doubt themselves and their power to undermine freedom," Zelenskyy cautioned. "When the free world hesitates, that's when dictatorships celebrate, and their most dangerous and vicious rape."

Last week, the Biden administration announced a $175 million package of military aid to Ukraine under the Presidential Drawdown Authority -- a fund that is running dry, which is why the administration is requesting more from Congress.

Kirby told reporters Monday he "fully expects" more security assistance for Ukraine to be announced before the end of the month.

ABC News' Luis Martinez and Allison Pecorin contributed to this story.