But it's another story in the House, where it wasn't winning a lot of fans among conservatives.
Rep. Joe Barton, R-Texas, signaled that conservative members of the House were deeply skeptical. He said any bill had to have serious spending cuts for him to vote to raise the debt ceiling and said he thought Obama and Treasury Secretary Jack Lew had more flexibility than they had said publicly.
"No deal is better than a bad deal," Barton said.
Asked whether the emerging package contained any victories for Republicans, Rep. James Lankford, R-Okla., a member of the House GOP leadership, said, "Not that I've seen so far, no."
McConnell briefed House Speaker John Boehner, R-Ohio, on Monday on the status of the Senate talks. Both House and Senate GOP leaders scheduled closed-door meetings of their respective memberships for Tuesday.
The stock market turned positive on bullish predictions from Reid and McConnell. The Dow Jones industrial average, which had risen strongly late last week on hopes of an agreement, rose another 64 points Monday.
In addition to approving legislation to fund the government until late this year and avert a possible debt crisis later this week or month, the potential pact would set up broader budget negotiations between the GOP-controlled House and Democratic-led Senate. One goal of those talks would be to ease automatic spending cuts that began in March and could deepen in January, when about $20 billion in further cuts are set to slam the Pentagon.
Democrats were standing against a GOP-backed proposal to suspend a medical device tax that was enacted as part of the health care law.
Democrats also were seeking to preserve the Treasury Department's ability to use extraordinary accounting measures to buy additional time after the government reaches any extended debt ceiling. Such measures have permitted Treasury to avert a default for almost five months since the government officially hit the debt limit in mid-May, but wouldn't buy anywhere near that kind of time next year, experts said.
Some lawmakers are frustrated that defusing the immediate standoff simply sets up another fight next year.
"It's punting because no one, Democrats, Republicans wants to face up — and the American people — to the tough reality that we're in," said Sen. Richard Shelby, R-Ala. "It's all a temporary fix."
Associated Press writers Donna Cassata, David Espo, Henry C. Jackson and Alan Fram contributed to this report.