A spokesman for Senator Harry Reid said the Senate majority leader has “signed off on the debt-ceiling agreement pending caucus approval.”

Mr. Reid, a Nevada Democrat, also raised the possibility that his chamber might vote as early as Sunday night on a yet-to-be-announced debt ceiling compromise designed to avert a potential economic crisis this week. When he emerged from a two-hour meeting with other Democratic lawmakers and was asked whether the Senate would vote on a deal Sunday.

“I hope so,” he told a swarm of reporters.

A Sunday vote seemed unlikely just a few hours earlier as top lawmakers and the White House continued to work behind closed doors to finalize a debt agreement that would cut spending by more than $2.5 trillion and raise the debt ceiling into 2013.

The Department of Treasury has said it will exhaust its borrowing authority at midnight on Tuesday night if the debt ceiling is not increased.

Voting quickly is not a hallmark of the Senate, where procedures often slow the legislative work to a crawl. For Mr. Reid to move to a vote by Sunday evening would require plenty of cooperation from the Republican minority to waive several of those procedures. But Democratic leadership aides said that they are “close” to a deal and that Senate Democrats are ready to present the plan to members of their caucus.

However, Democratic officials said that one snag remained Sunday evening involving the extent to which the agreement would make cuts in defense spending.

House Speaker John A. Boehner was attempting to scale back the $350 billion in immediate Pentagon cuts that would be included in the initial $1 trillion in spending cuts. Democrats said Mr. Boehner was also trying to minimize the amount of automatic cuts to defense spending that would occur if a special congressional committee was unable to reach a broader agreement later this year.

Senator Mitch McConnell of Kentucky, the Republican leader, has already agreed to the defense cuts, according to Democratic leadership aides. Democratic lawmakers were urging the White House to stand firm against Mr. Boehner’s demands, they said.