Mr Trump, a Republican, claimed to have won and vowed to launch a Supreme Court challenge, baselessly alleging fraud.
Earlier Mr Biden, a Democrat, said he was "on track" to victory.
Millions of votes remain uncounted and no candidate can credibly claim victory as yet. There is no evidence of fraud.
The US is on course for the highest electoral turnout in a century. More than 100 million people cast their ballots in early voting before election day, and tens of millions more added their vote on Tuesday.
With the nation on edge, the final result may not be known for days.
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