The duration of a debt limit increase is also emerging as a line in the sand, with the White House insisting on a higher increase than Republicans have floated. Both sides could agree to raise the limit for only a couple of months as they seek to finish budget negotiations. But Mr. Biden’s aides want to avoid such a short-term solution and do not want to conduct an entirely new round of negotiations next year. As a result, any larger budget deal would likely need to raise the limit for borrowing needs past the next presidential election, instead of into early next year, as the Republican bill did.

Republicans acknowledge that the White House has laid down numerous red lines but say that the president will have to relent in some areas if an agreement is to be struck.

“None of us, nobody in this room, thinks Joe Biden will get everything he wants in this deal,” said Mr. Johnson. “That means by definition he will have to accept a number of things he says he refuses to accept.”

“We’re not going to negotiate with ourselves,” said Representative Garret Graves, a Louisiana Republican deputized by Mr. McCarthy to shepherd Republicans through the debt ceiling showdown. “We’re going to have substantial savings moving forward.”

Biden administration officials are also open to striking a deal with Republicans on accelerating permitting for a wide range of energy projects, including wind, oil, gas and solar — a top priority of Senator Joe Manchin III, Democrat of West Virginia.