A bipartisan group huddled for hours on Sunday trying to end the standoff.
"Encouraged that 22 members from both parties attended," Republican Senator Susan Collins said in a tweet.
A member of that group, Republican Senator Lindsey Graham, told reporters that McConnell has "included immigration in the mix for the first time," addressing an issue at the heart of the dispute, if funding is provided until February 8.
"Mitch has got a proposal. We should rally around that," said Graham, blaming "White House staff" for not working with lawmakers.
"I'm begging the White House to find a way to work with us," Graham said.
Yet, rank-and-file Republicans expressed skepticism over the bipartisan effort.
"I don't see any of our people interested in some half-baked idea that's produced by a self-appointed group of senators," Oklahoma Republican House member Tom Cole said.
Lawmakers have traded bitter recriminations for the failure to pass a stop-gap funding measure, and McConnell once again sought to blame Senate Minority Leader Chuck Schumer.
Trump early Sunday encouraged the Senate's Republican leaders to invoke the "nuclear option" -- a procedural maneuver to change the chamber's rules to allow passage of a budget by a simple majority of 51 votes to end the shutdown.
But Senate leaders have been wary of such a move in the past, as it could come back to haunt them the next time the other party holds a majority.
White House press secretary Sarah Sanders said Trump had spoken during the day with McConnell and Senate Majority Whip John Cornyn. She did not mention Trump's speaking with any Democrats but said White House director of legislative affairs Marc Short had been in touch with members of both parties and updated the president.
"We are continuing to work hard towards reopening the government," she said.
Essential services continue
At the heart of the dispute is the issue of undocumented immigration.
Democrats have accused Republicans of poisoning chances of a deal and pandering to Trump's populist base by refusing to back a program that protects an estimated 700,000 "Dreamers" -- undocumented immigrants who arrived as children -- from deportation.
Schumer said he and Democrats were willing to compromise, but Trump "can't take yes for an answer -- that's why we're here."
"I'm willing to seal the deal, to sit and work right now with the president or anyone he designates -- let's get it done," Schumer said.
Trump has said Democrats are "far more concerned with Illegal Immigrants than they are with our great Military or Safety at our dangerous Southern Border."
Essential federal services and military activity are continuing, but even active-duty troops will not be paid until a deal is reached to reopen the US government.
There have been four government shutdowns since 1990. In the last one, in 2013, more than 800,000 government workers were put on temporary leave.
"We're just in a holding pattern. We just have to wait and see. It's scary," Noelle Joll, 50, a furloughed US government employee, told AFP in Washington.
New York Governor Andrew Cuomo, a Democrat, said Sunday that state funding would pay for the reopening Monday of the Statue of Liberty, which was among facilities affected by the shutdown.
A deal had appeared likely on Friday afternoon, when Trump -- who has touted himself as a master negotiator -- seemed to be close to an agreement with Schumer on protecting Dreamers.