He and Mr. McConnell are now back leading a negotiations process that has so far been marred by mistrust and anger.
Mr. Reid was careful not to sound too hopeful but said: "Senator McConnell and I have been in this body a long time. We've done things for a long time together. I know him. He knows me. We don't agree on everything, and that's, as you know, an understatement."
He added, "This is what legislating is all about."
For Speaker John A. Boehner and House Republicans, the options were much grimmer. The speaker, who told his members in an early morning meeting that his efforts to strike a compromise with the White House had failed, can continue to push back against President Obama and hope for some give in the White House's stance that it will not negotiate until the government is reopened and the debt ceiling is raised. If he were to pursue a more accommodating approach similar to his Republican colleagues in the Senate, he risks angering the conservatives who dominate his conference.
Those conservatives left their meeting at the Capitol on Saturday in a sour mood, with many saying they were outraged that Mr. Obama still refused to meet them halfway.
(Read more: Shutdown's toll: Idled research to closed wallets)
Representative John Carter of Texas described Mr. Obama as "acting like a royal president."
"He's still 'my way or the highway,' " Mr. Carter said.
With concerns growing that global financial markets could be thrown into turmoil if Congress does not agree to raise the debt ceiling, Republicans said they did not know whether Mr. Boehner would have enough support from the most conservative members in his conference to put a Senate plan up for a vote — if the leaders reach a deal.
"The question is: What will Senate Republicans do, what will Senate Democrats do?" said Representative Adam Kinzinger, Republican of Illinois.
Many Republicans said that however frustrated they were that the White House would not negotiate with them, they were just as dismayed with House colleagues who would not back down from their demands that any deal include provisions to chip away at the health care law.
"The problem here is that we don't have a functioning majority," said Representative Devin Nunes, Republican of California. "After three weeks of this, they're still not figuring it out. I don't know what it takes."
The proposal House Republicans presented to the White House late last week called for increasing the Treasury Department's authority to borrow money through Nov. 22, but only if Mr. Obama agreed to more expansive talks about overhauling the budget.
The failure of talks with the White House further strained the relationship between the president and House Republicans, whose refusal to approve a spending bill until less it stripped financing from the health care law shut down the government
Representative Aaron Schock, Republican of Illinois, there had been "a total breakdown in trust" between House Republicans and the administration.
(Read more: The best ways to punish Congress)
"You don't tell the speaker, the majority leader, the majority whip, 'We're going to negotiate.' Then they come and tell our entire conference, 'We're going to negotiate,' " he said. "And then 24 hours later, you recant."
Feelings ran so high on the House floor on Saturday morning that there was a brief altercation between Representative Joseph Crowley, a Democrat from New York City, and Chris Vieson, the floor director for Representative Eric Cantor of Virginia, the majority leader. There were conflicting reports about whether the conflict became physical or was confined to words, but both sides said they had apologized.
If Republicans needed any reminder about how outraged their most conservative supporters would be if they committed to a compromise that did not include provisions to weaken the health care law, they needed look no further than out the window. Glenn Beck, the fiery radio personality, was leading a group of Tea Party activists on the National Mall.
As the House met on yet another proposal that would go nowhere in the Senate, Mr. Reid called a vote to begin debate on a Democratic proposal that would extend the debt ceiling through the end of 2014 with no strings attached. No Senate Republicans voted yes, and the measure failed to reach the 60-vote threshold it needed.
Despite the expected failure of the Senate Democrats' proposal, Senator Harry Reid, the majority leader, could execute a procedural maneuver known as a motion to recommit that would allow him to bring the plan to a vote again, and aides indicated that he would to keep pressure on Republicans as the debt ceiling deadline neared on Thursday.
The White House called the Senate Republicans' action irresponsible. "This bill would have taken the threat of default off the table and given our nation's businesses and the economy the certainty we need," said Jay Carney, the White House press secretary. "With five days until the government runs out of borrowing authority, Congress needs to move forward with a solution."
Senator Charles E. Schumer, Democrat of New York, said the Republicans' rejection of the debt-limit bill "is playing with fire."
House Republicans remained skeptical that any Senate compromise could pass muster in the Republican-controlled House. Representative Thomas Massie, Republican of Kentucky, said simply: "Senate Republicans don't run the Senate. So we're not taking our lead from them."
And despite encouraging signs from Mr. McConnell and Mr. Reid, senators from both parties said they did not have unrealistic expectations of any quick solution. "Let's be honest where we are," said Senator Richard J. Durbin of Illinois, the No. 2 Senate Democrat.