Rarely used in US Congressional proceedings, the nuclear or constitutional option is a parliamentary procedure that allows the US Senate to override a rule or precedent by majority vote.
However, the Republicans are unlikely to opt for this, Senate Majority Leader Mitch McConnell said.
Hundreds of thousands of federal government employees would not return to work tomorrow as the Senate -- the upper chamber of the United States Congress -- did not pass a legislation required to fund the expenses of the federal government beginning yesterday due to opposition from the Democratic Senators.
In a procedural vote to fund the government, even for a short-term four-week period till February 16, the Republican last night got 51 votes as against 60 required to move ahead with the legislation.
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"Great to see how hard Republicans are fighting for our military and safety at the border. The Democrats just want illegal immigrants to pour into our nation unchecked. If stalemate continues, Republicans should go to 51 per cent (nuclear-option) and vote on real, long-term budget, no CR's!" Trump said in a tweet today.
"The Republican Conference opposes changing the rules on legislation," his spokesman said.
The issue had come up during internal party deliberations.
Both the Democratic and Republican leaders in the Senate said that they were continuing with their negotiations and short out their differences so as to end the government shutdown.
But the White House alleged that the Democrats want to prolong it for days or at least till Trump delivers his State of the Union address on January 30.
"It could go several days because I think there's other Democrats who want to see the president give the State of the Union during a shutdown," the Director of Office of Management of Budget Mick Mulvaney told Fox News.
"You have to ask the Senate Democrats. They could open this today if they wanted to," Mulvaney said.
The Democrats were quick in opposing the nuclear option.
"This is another example of President Trump growing a tweet in the middle of bipartisan negotiations that are making progress," Democratic Senator Chris Coons told the TV channel.
Ending the government shutdown is president's responsibility, he said.
"He should, instead of throwing tweets from the White House, pull together the four leaders of the House and the Senate on a bipartisan basis today and negotiate.
Top Democratic leader Dick Durbin refused to say when the shutdown would end.
Meanwhile the White House, on its public telephone and general email, has put an auto-responder which alleges that the Democrats have put the government to hostage.
"Thank you for calling the White House. Unfortunately, we cannot answer your call today because congressional Democrats are holding government funding, including funding for our troops and other national security priorities, hostage to an unrelated immigration debate," says a woman's voice when one calls the White House general number.
"Due to this obstruction, the government is shutdown. In the meantime, you can leave a comment for the President. We look forward to taking your calls as soon as the government re-opens," it said.
"Unfortunately, I am out of the office today because congressional Democrats are holding government funding -- including funding for our troops and other national security priorities -- hostage to an unrelated immigration debate," one of those emails said.