Negotiations are continuing to unfold in an attempt to reach a debt limit deal, but major differences between House Republicans and the White House have yet to be bridged, and the pressure is only intensifying as the risk of default grows ever more real with each day.

Republicans have long said that spending cuts must be paired with any increase in the debt limit -- an issue that is proving to be the central sticking point as Democrats argue the cuts Republicans want are too extreme, though the White House has expressed a willingness to cut some spending.

Asked if there is any general agreements on cuts, GOP Rep. Garret Graves, who has served as a chief negotiator during the talks, said on Tuesday, "No, that's our biggest gap."

Graves made clear that a wide array of significant differences remain, even as the timeline to avoid default grows shorter.

"Look, there are some big bright red lines on both sides," he said. "We do not have any of those issues closed out."

McCarthy tells conference: 'We are nowhere near a deal'

Underscoring just how far apart the two sides are, House Speaker Kevin McCarthy told Republicans during a closed-door meeting on Tuesday, "We are nowhere near a deal," according to sources in the room.

The window to secure a deal is rapidly closing and the stakes are incredibly high with the Treasury Department continuing to say the US could default by June 1. A first-ever default for the US would likely trigger a global economic catastrophe.

McCarthy met with President Joe Biden at the White House on Monday, a meeting the speaker and the president both said was "productive," but that did not yield a breakthrough in negotiations or end in a deal.

Congressional Democrats, meanwhile, continue to push back against positions Republicans have staked out in negotiations and express heightened concern over the ticking clock.

House Democratic Caucus Chair Pete Aguilar argued that Republicans have been pushed to take extreme positions by far-right members of the GOP conference.

"This is tough, this is not where we should be. Speaker McCarthy is being held captive by the Freedom Caucus," Aguilar told reporters on Tuesday.

"We're concerned that Republicans are not only moving the goalposts, but continue to hold onto the most extreme elements of their proposals," he said.

McCarthy's position: Republicans won't make concessions

McCarthy, despite saying the two sides are still far apart, said Tuesday he thinks it is possible to get everything done by the June 1 deadline. "We could still finish this by June 1st," he told reporters.

But in a sign there may not be much room to maneuver in negotiations, McCarthy told CNN's Manu Raju, "We're going to raise the debt ceiling," when asked what concessions he would make -- a significant remark that indicates Republicans are not willing to give any more than raising the debt ceiling in exchange for their demands.

When Raju pressed the speaker, asking if that is his only concession, McCarthy said, "Everything we're going to do is going to make America stronger."

McCarthy's comment Tuesday that raising the debt ceiling is the only concession he will make rankled a White House that just a day earlier viewed conversations with McCarthy as productive, according to two sources familiar with the negotiations.

A Democratic official slammed McCarthy for his comments, accusing him of refusing to compromise and being beholden to the most conservative members of his caucus as negotiations languish.

The official slammed McCarthy's demand that defense spending increase while non-defense spending decrease as extreme and out of step with budget deals over the last decade.

What's happens when there's a deal?

Another significant challenge facing negotiators is that even if a deal is reached, that is far from the end of the road to prevent default.

Legislative text will need to be written, which can be arduous and complicated work as lawmakers and staff dive into nitty-gritty policy details -- a part of the process toward final passage of any bill that can often lead to further issues and eleventh-hour hangups over disagreements about the fine print.

Then, leaders from both parties will need to wrangle the votes to pass a bill, no small task with narrow majorities in both chambers. After that, a deal would need to be brought to the floor, a process that can take days to play out in both chambers, though there are mechanisms available to leadership to speed things up.