Hundreds of thousands of federal staff face the prospect of unpaid leave.
On Saturday, recriminations flew around over the Senate's failure to pass a new budget and prevent the shutdown.
A bill to fund the federal government for the coming weeks did not receive the required 60 votes by Friday.
The Republican leader of the US Senate, Mitch McConnell, has said there will be a vote at 01:00 in the early hours of Monday (06:00 GMT) on a bill to fund the government until 8 February.
The last government shutdown was in 2013, and lasted for 16 days.
