No candidate can credibly claim to have won as yet, and both campaigns said they had potential paths to victory.
Mr Biden's campaign said the race was "moving to a conclusion in our favour".
But Mr Trump, a Republican, claimed to have won and vowed to launch a Supreme Court challenge, without providing evidence of fraud.
Several key states are expected to finish counting by the end of Wednesday but the election may not be decided for days.
The Biden campaign said it expected to win because several states would be called for them on Wednesday or Thursday, but the Trump campaign said it was confident that the maths was in its favour.
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.