Notably the House of Representative has already approved a short-term extension of government funding for about a month, but the legislation is now stuck in the Senate.
Eliminating the 60-vote threshold to break a legislative filibuster would remove significant powers for the minority party in the Senate, and party leaders have been reluctant to do so in the past because of the consequences it would pose when their party returns to the minority.
The Senate on Sunday delayed a vote to end the government shutdown that was scheduled for 1 a.m. on Monday morning.
"It all really stems from the president, whose inability to clinch a deal has created the Trump shutdown", Schumer said. "It takes 60 votes in the Senate". GOP Arizona Sen. Jeff Flake, another previous no-vote, announced he would vote in favor of reopening the government Monday.
The movement comes thanks in part to commitment from McConnell and other Republicans in bipartisan meetings, according to four Democratic sources.
Chief of Staff John Kelly has spoken with House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell. "This is his shutdown", Democratic Senator Dick Durbin told the CBS network on Saturday.
But Congress has often failed to meet this deadline and negotiations continue well into the new year, with the previous year's funding to federal agencies extended on a temporary basis.
The shutdown began on the first anniversary of President Trump's inauguration.
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.
Democrats have been seeking a deal to protect the "Dreamers", who have been shielded against deportation by the Deferred Action for Childhood Arrivals program, or DACA, which Trump halted a year ago.
Among those losing pay until the impasse is solved are more than 300,000 civilian defence workers.
But Democratic Sen. Dick Durbin of IL says that would mean the end of the Senate as the Founding Fathers envisioned it.
"They need to open the government tonight or tomorrow and then we can start talking", Mulvaney said on CNN.
Early Saturday morning, Senate Minority Leader Chuck Schumer, D-N.Y., said he had offered a possible solution that Democrats would support, but it was rejected by the White House. "But rather than solve problems, Democratic leadership preferred a shutdown that has unsafe consequences for our national defense".
Giving an insight into the planning goal, Mulvaney said the military will still go to work; the border will still be patrolled; fire folks will still be fighting the fires; and the parks will be open. "And so, really, I'd say, let's vote on it. Let's just put it forward and vote on it".
After the government shut down this weekend, all nonessential federal services were suspended.
The shutdown began Saturday after Democrats derailed a Republican measure that would have kept government open until February 16.
Behind the scenes, some lawmakers continued to discuss a short-term stopgap bill to give more time for talks on immigration legislation sought by Democrats.
The White House issued a statement blaming Senate Democrats for blocking a bill to avert a shutdown and said it would not negotiate on immigration, a key demand of Democrats. Lawmakers said they were taking the proposal to leadership Sunday afternoon.
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