A bill to fund the federal government for the coming weeks did not receive the required 60 votes by the deadline of midnight on Friday.
President Trump accused the Democrats of putting politics above the interests of the American people.
The Democrats blame him for rejecting bipartisan compromise proposals.
Both the Republican and Democrat congressional leaders say they will keep talking, and the White House budget chief has expressed optimism that a resolution will be found before Monday.
But if not, hundreds of thousands of federal workers face the prospect of no work and shuttered offices at the start of the working week.
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 54 seat majority in the Senate, the Republicans did not have enough seats 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 to the bill 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”.
“As soon as you take one step forward, the hard-right forces the president three-steps back,” Mr Schumer said.
White House budget director Mick Mulvaney hit back, accusing Mr Schumer of “mischaracterising” his meeting with the president on Friday, and saying the senator should be “more honest with the president of the United States”.
Negotiations in both houses of Congress are continuing on Saturday, and Democrats and Republicans were also due to hold separate meetings.
White House spokeswoman Sarah Sanders warned: “The president will not negotiate on immigration reform until Democrats stop playing games and reopen the government.”
The US budget must be approved by 1 October – the start of the federal financial year.
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.
Because Congress failed to agree an extension that would have maintained government funding through to 16 February, it means many federal agencies effectively closed for business as of 00:01 Saturday (05:01 GMT). (Courtesy BBC)