Mr. McConnell reiterated a pledge he offered on Sunday night that he intended for the Senate to take up immigration legislation in February if the issue has not been resolved by then. A major question as senators seek to end the shutdown is what kind of commitment Mr. McConnell is willing to make regarding the consideration of legislation for the young immigrants, a central issue in the current impasse.

On Monday, Mr. McConnell pledged that the Senate’s immigration debate would have “a level playing field at the outset and an amendment process that is fair to all sides.”

“The very first step is ending the government shutdown,” he said.

— Thomas Kaplan

Democrats wanted more than McConnell’s word.

Moderate Senate Democrats Monday morning had sought a firmer commitment from Mr. McConnell that the Senate would move to address the fate of hundreds of thousands of young undocumented immigrants, known as Dreamers, in the coming weeks.

The Democrats were part of a bipartisan group of more than 20 senators working throughout the weekend to forge a compromise to reopen the government. Mr. McConnell signaled Sunday night that he was listening to their demands, saying from the Senate floor that he intended to move ahead with immigration legislation in February if the issue had not been resolved by then.

But on Monday, Democrats wanted more in exchange for the votes to end the shutdown.

“Well I think the first thing he needs to do is strengthen his statement from last night,” said Senator Angus King of Maine, an independent who votes with the Democrats. “ ‘I intend.’ I would much rather he say, ‘I commit’ or ‘I will move.’”

As senators from the group shuffled in and out of leadership offices, Senator Jeff Flake, Republican of Arizona and another member of the group, expressed optimism that such a public statement by Mr. McConnell would be enough to win over enough Democrats to vote to end the shutdown. Some Democrats called on Mr. McConnell to delay a procedural vote schedule for noon.

The crux issue, it seemed, was whether the majority leader could be trusted to keep his word. Democrats have not forgiven Mr. McConnell for blocking the Supreme Court nomination of Merrick Garland for almost a year pending the election of a Republican to the White House. And Mr. McConnell’s promises to Republican Senators Flake and Susan Collins of Maine for votes on health care and immigration in exchange for their support of the tax cut have yet to materialize.