The deal may be imperiled by last-minute reluctance from Boehner, who worries it cuts too much from future Pentagon budgets.

Sen. James Inhofe (R-Okla.) missed Sunday’s procedural vote on Senate Majority Harry Reid’s (D-Nev.) deficit reduction proposal because he was attending an aviation celebration in Oshkosh, Wisconsin, his communications director told The Hill.

“Senator Inhofe missed today's procedural vote, because he had prior commitments at AirVenture in Oshkosh, WI, including meeting with and helping those seeking to strengthen the aviation sector of Oklahoma's economy,” Inhofe communications director Jared Young wrote in an email.

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When asked if a Sunday vote was possible on a White House, GOP compromise Sen. Reid said, "I hope so."

Rep. Raul Grijalva, co-chair of the Congressional Progressive Caucus, called the possible compromise a “cure as bad as the disease.”

Senate Majority Leader Harry Reid (D-Nev.) instructed senators to stay close to the Capitol Sunday afternoon as congressional leaders and President Obama furiously negotiated a deal to raise the nation's debt ceiling.. 

“I would not suggest [going to] a ball game,” said Reid from the floor to some laughter. “Maybe closer than that.”

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House leaders in both parties will have a tough fight passing a debt-ceiling package being finalized by Obama and Republicans.

Senate Majority Leader Reid joined his Republican counterpart Sunday in expressing optimism on a debt deal currently in the works.

The Senate meets at noon, after having adjourned Saturday night without taking a vote on a plan from Senate Majority Leader Harry Reid (D-Nev.) to raise the debt ceiling.

Senate passage of Reid's plan or any plan has always been in doubt, and word of a new deal between the White House and congressional Republicans may let the Senate escape a series of failed votes.

A congressional source said President Obama and GOP leaders had reached a tentative debt-ceiling deal; Reid says details unresolved.