WASHINGTON (Reuters) - Republican and Democratic leaders of the U.S. Senate held talks on Sunday seeking to break an impasse that has kept the U.S. government shut down for two days, but it was unclear if a deal could be struck to reopen federal agencies by the start of the work week.

The high-level meeting followed discussions among a bipartisan group of moderate senators who said they made progress but offered few details of a possible compromise. The ultimate decision was up to Senate Republican Leader Mitch McConnell and his Democratic counterpart, Senator Chuck Schumer, they said.

An emerging deal, kick-started by the meeting of about two dozen Republican and Democratic centrists, appeared to involve reopening the government by Monday morning in return for a promise by majority Republicans to deal with Democrats' immigration concerns in coming weeks.

“I’m hoping they are going to agree to something soon," Republican Senator Jeff Flake said of the McConnell-Schumer talks. Roy Blunt, a junior member of the Republican Senate leadership, said there was a "chance" of a resolution on Sunday night.

Funding for federal agencies ran out at midnight on Friday, and was not renewed amid a dispute between U.S. President Donald Trump and Democrats over immigration.

Refusing to support another short-term government funding extension last week, Democrats demanded that Trump live up to an earlier agreement to protect "Dreamers," immigrants brought to the United States illegally as children, from deportation.

"We need to have a substantive answer, and the only person who can lead us to that is President Trump. This is his shutdown," Dick Durbin, the second-ranking Democrat in the Senate, said on the CBS "Face the Nation" program.

Trump and other Republicans were just as adamant on Sunday, saying they would not negotiate immigration or other issues as long as all but essential government services remain shuttered.

"We will not negotiate on the status of unlawful immigrants while Senator Schumer and the Democrats hold the government for millions of Americans and our troops hostage," White House press secretary Sarah Sanders said.

The shutdown is the first since a 16-day closure in October 2013. Its effects will likely be more visible on Monday, when financial markets and federal offices open.

The Senate will vote at 1 a.m. (0600 GMT) on Monday on whether to advance a measure to fund the government through Feb. 8, McConnell said on Saturday. But several senators said on Sunday the vote could be accelerated.

"We’re going to open the government and solve immigration at the same time," said Republican Senator Lindsey Graham.

Trump, who spoke by phone to Republican lawmakers and agency heads on Sunday but has not had any public appearances since Friday, canceled a weekend trip to his Mar-a-Lago estate in Florida that included a major fundraiser on the anniversary of his first year as president.

The White House said his planned trip to the World Economic Forum in Davos, Switzerland, next week was in flux because of the standoff.

With elections set for November for a third of U.S. Senate seats and the entire House of Representatives, both sides are maneuvering to blame the other for the shutdown.

Trump said earlier on Sunday that if the stalemate continued, Republicans should change Senate rules so a measure could be passed to fund the government.

Current Senate rules require a super-majority of three-fifths of the chamber, usually 60 out of 100, for legislation to clear procedural hurdles and pass.

"If stalemate continues, Republicans should go to 51 percent (Nuclear Option) and vote on real, long term budget," Trump said on Twitter.

But McConnell rejected the idea.

Republicans hold a slim 51-49 majority in the Senate.

'HOSTAGES RIPE FOR THE TAKING'

In a Senate floor speech on Sunday, McConnell accused Schumer of imperiling children's health care, military training, veterans' care and other programs.

"To most Americans, those sound like fundamental responsibilities" of government, McConnell said. "To the Democratic leader, apparently they sound like hostages ripe for the taking."

White House budget director Mick Mulvaney said Trump had instructed him to ease the effects of the shutdown as much as possible.