Originally Published: 01 FEB 23 05:00 ET

Updated: 01 FEB 23 17:32 ET

By Maegan Vazquez and Clare Foran, CNN

(CNN) -- House Speaker Kevin McCarthy did not walk away from his highly anticipated White House meeting on Wednesday with an agreement in hand to address the debt limit, but signaled optimism that both he and President Joe Biden can reach consensus "long before" the United States reaches default.

McCarthy and the president exchanged political jabs ahead of the meeting -- preempting their negotiations with red lines relayed to the press and on social media. But emerging from the West Wing on Wednesday, the new House speaker had an unexpectedly hopeful tone as he underscored that he believes that they can come to an agreement.

McCarthy called it "a good first meeting," adding, "We both have different perspectives on this, but I thought this was a good meeting. We promised we would continue the conversation and we'll see if we can get there. I think that at the end of the day, we can find common ground."

"I told the president I would like to see if we can come to an agreement long before the deadline and we can start working on other things," McCarthy added in remarks outside the White House.

Following his first White House meeting since he won the speakership, McCarthy said he believes that a funding agreement could be reached for the next two years and that "you won't see omnibuses anymore."

"You'll see the Senate and the House actually do what the American public has elected them to do," he added.

The highly anticipated meeting was expected to influence how the fight to raise the national debt limit unfolds as the White House and the new House GOP majority are at odds over how to resolve the critical issue.

House Republicans say that lifting the borrowing cap must be tied to spending reductions. The White House, however, has countered that it will not offer concessions or negotiate on raising the debt ceiling.

The US hit the debt ceiling set by Congress in January, forcing the Treasury Department to start taking extraordinary measures to keep the government paying its bills and escalating pressure on Capitol Hill to avoid a catastrophic default.

The debt limit fight will be an early test of McCarthy's leadership as House speaker, where he has to balance competing demands from different factions of his conference amid a razor-thin majority. It will also shed light on how, and to what extent, McCarthy and Biden are able to work with one another.

Senate Republicans have indicated they will sit back and see how the House GOP maneuvers a way to raise the $31.4 trillion borrowing limit -- before deciding if they need to insert themselves into the process.

McCarthy told reporters on Tuesday the nation has reached "a critical point" with respect to the debt limit.

Battle lines begin to be drawn

Republicans face a political risk as they push to cut spending: If they propose cuts to popular government programs and services, they could face a public backlash.

While McCarthy had not settled on any individual proposal ahead of the Biden meeting and was unlikely to make a specific offer, he had heard suggestions from key players in his conference.

Ahead of Wednesday's meeting, McCarthy has been involved in extensive preparations, consulting regularly with allies on and off the Hill including former House Speaker Newt Gingrich as well as his relevant committee chairs who he has been leaning on for their policy expertise, such as Ways and Means Chairman Jason Smith of Missouri and Financial Services Committee Chairman Patrick McHenry of North Carolina, sources familiar with the preparation told CNN.

McCarthy and his House GOP allies are hashing out initial demands, discussing steep cuts to domestic programs and a trim to defense spending -- all the while steering clear of two programs to avoid voter blowback: Medicare and Social Security.

House Republicans had been hoping to strengthen their negotiating hand with the White House by uniting around a proposal, but finding conference-wide consensus on spending cuts has proved challenging.

The view from Republicans heading into Wednesday's meeting was that it is still early and there are still months of negotiations ahead -- meaning there's plenty of time for McCarthy to lay out specifics. Still, leaders have also been aware they have to begin laying the groundwork with their members now.

The White House, meanwhile, has continued to emphasize the critical importance of avoiding a catastrophic default.

McCarthy's position that cuts to Medicare and Social Security are not on the table in exchange for a debt ceiling increase has drawn skepticism the White House. And when asked for his message to McCarthy in the meeting, the president told CNN, "Show me your budget and I'll show you mine."

A White House spokesperson told CNN that Biden would remind McCarthy of his "Constitutional obligation to prevent a national default, as every other House and Senate leader in U.S. history has done, and as Leaders (Mitch) McConnell, (Chuck) Schumer, and (Hakeem) Jeffries have pledged to do."

"He will underscore that the economic security of all Americans cannot be held hostage to force unpopular cuts on working families," they added.

In a memo to "interested parties" dated Monday that was written by National Economic Council Director Brian Deese and Office of Management and Budget Director Shalanda Young, Biden's top economic advisers said the president intended to pose two questions to McCarthy on Wednesday: Whether McCarthy will commit to the US not defaulting on its financial obligations and when McCarthy and House Republicans will release their budget.

Biden, the officials wrote, "will seek a clear commitment from Speaker McCarthy that default -- as well as proposals from members of his Caucus for default by another name -- is unacceptable."

They added, "President Biden will ask Speaker McCarthy to publicly assure the American people and the rest of the world that the United States will, as always, honor all of its financial obligations."

A day ahead of the meeting, the president suggested McCarthy was entering the talks from a weakened position, hampered by agreements he made with an unruly GOP conference.

Calling McCarthy a "decent man," Biden nonetheless said he had been forced to cater to extremist Republicans in his quest to become speaker.

Biden said at a high-dollar fundraiser in Manhattan that McCarthy had to make commitments "that are just absolutely off the wall for the speaker of the House to make."

Responding to the president's fundraiser comments, McCarthy told reporters, "Apparently, he doesn't understand ... I'm looking forward to sitting down with the president, negotiating for the American public, the people of America, on how we can find savings. We've watched what the spending has done, we watched it brought us inflation, we watched the challenge that it happened. We're looking forward to changing the course."

Illinois Sen. Dick Durbin, the Senate majority whip, told "CNN This Morning" on Wednesday that Biden should "absolutely not" negotiate on raising the nation's borrowing limit and raised fears a default could tank the US economy. He reiterated his support for Biden's position, while leaving the door open for spending cuts during future negotiations.

House GOP Whip Tom Emmer said he doubted there would be any agreement on Wednesday, but said,"Everything's on the table" when asked about defense spending cuts.

He also said he expected McCarthy to reassure the president that there will not be a default and that spending cuts for Social Security and Medicare will not be considered.