The US Senate has failed to reach a temporary budget deal to keep it open.
Some employees will not be paid until the stalemate is resolved.
A vote will take place at noon in Washington (17:00 GMT) on whether to re-open government. Immigration remains one of the main sticking points between Republicans and Democrats.
Democrats have refused to back a temporary budget deal until their concerns on immigration reform are dealt with.
Essential federal services are still running across the country, but non-essential workers are not required to report to work.
It missed a deadline. At midnight on Friday, lawmakers failed to agree on a spending bill. The bill was not a plan for funding for the whole of 2018, but would have kept things running until the middle of next month.
Efforts to reach a compromise ahead of the working week failed in a rare Senate session late on Sunday.
A vote to end the shutdown was postponed until midday on Monday, meaning many federal government offices will not open as the shutdown enters its third day.
Under Senate rules, the bill needs 60 votes in the 100-member chamber.
The Republicans currently have 51 senators, so they need some Democratic support to pass a budget.
Democrats want President Trump to negotiate over immigration as part of a budget deal, but Republicans say no agreement is possible while federal government services are closed.
Republicans want funding for border security – including a proposed border wall with Mexico – and immigration reforms, as well as increased military spending.
On Saturday, Mr Trump said the “nuclear option” of a simple majority vote was necessary to end the impasse.
It means no pay for those federal employees who are “furloughed” – on unpaid leave – even though their workplaces are not open.
The last government shutdown was in 2013, and lasted for 16 days.
It cost the government $2bn in lost productivity and led to “significant negative effects on the economy”, the OMB said at the time.
After the 2013 shutdown, Congress passed a bill ensuring employees who were on unpaid leave received pay cheques for the time they missed. However, receiving pay is not guaranteed this time as Congress would have to approve it again.
The shutdown began on the first anniversary of President Trump’s inauguration. His trip to the World Economic Forum in Davos, Switzerland, has also been called into question.
Most staff in the departments of housing, environment, education and commerce will be staying at home on Monday. Half of workers in the treasury, health, defence and transportation departments will also not be going to work.
But essential services that protect “life or human property” will continue, including national security, postal services, air traffic control, some medical services, disaster assistance, prisons, taxation and electricity generation.
And the Trump administration said it planned to keep national parks open – their closure in the 2013 shutdown provoked an angry public reaction.
However, many rangers and staffers will not go into work.
The Statue of Liberty was closed on Sunday but the governor of New York said he will use state funds to keep the historic landmark open from Monday.
However, Independence Hall which is home to the Liberty Bell in Philadelphia, will be closed along with other national landmarks.
The defence department issued guidance saying that all military active duty personnel will continue to work without pay.
Other closures include:
This is the first time a government shutdown has happened while one party, the Republicans, controls both Congress and the White House.
Friday’s vote fell 50-49, far short of the 60 needed to advance the bill. This is due to a number of key disagreements.
Democrats have demanded protection from deportation of more than 700,000 undocumented immigrants who entered the US as children.
“I hope it is just a matter of hours or days. But we need to have a substantive answer, and the only person who can lead us to that is President Trump. This is his shutdown,” Democratic Senator Dick Durbin told the CBS network on Saturday.
But Vice-President Mike Pence reiterated his party’s stance in a speech to US troops in the Middle East on Sunday.
“We’re not going to reopen negotiations on illegal immigration until they reopen the government and give you, our soldiers and your families, the benefits and wages you’ve earned,” he said.