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.
This is the first time a government shutdown has happened while one party, the Republicans, controls both Congress and the White House.
The vote on Friday was 50-49, falling far short of the 60 needed to advance the bill. With a 51-seat majority in the Senate, the Republicans do not have enough votes to pass the bill without some support from the Democrats.
They want funding for border security - including the border wall - and immigration reforms, as well as increased military spending.
The Democrats have demanded protection from deportation of more than 700,000 undocumented immigrants who entered the US as children.
The Republicans added a sweetener in the form of a six-year extension to a health insurance programme for children in lower-income families. But Democrats want this programme extended permanently.
Mr Trump accused the Democrats of being "far more concerned with illegal immigrants than they are with our great military or safety at our dangerous southern border".
But the leading Senate Democrat, Chuck Schumer, blamed the president, saying Mr Trump was under pressure from "hard-right forces within the administration".
White House spokeswoman Sarah Sanders warned: "The president will not negotiate on immigration reform until Democrats stop playing games and reopen the government."