Biden made clear on Friday that any nomination should await the results of the election.
"The voters should pick the president and the president should pick the justice for the Senate to consider," he said.
Senate Democratic leader Chuck Schumer said that "this vacancy should not be filled until we have a new president" - carefully echoing the words of Senate Republican leader Mitch McConnell in 2016 when he blocked Obama nominee Merrick Garland.
Republicans in theory have the Senate votes to push through a Trump nominee, but they could be blocked by only a handful of defections.
Analysts predicted the Democrats would do their best to drag out the process while fanning public outrage over what Democrats called the Republicans' hypocrisy.
"Their option is to build a groundswell (of public opposition)... to try to convince at least four Republican senators to vote no on whoever the president puts forward," Amy Howe, co-founder of a Supreme Court blog, said on CNN.
A handful of moderate Republican senators - including Lisa Murkowski of Alaska and Susan Collins of Maine - have already expressed doubts about a rushed vote.
Other Republicans facing tough re-election races in closely divided states may be loath to take a stand before the election.
Democrats are sure to use the past words of some Republicans against them.
Senator Lindsey Graham, a close Republican ally of Trump's, said in 2018, that "if an opening comes in the last year of President Trump's term, and the primary process has started, we'll wait to the next election."
Graham chairs the Senate judiciary committee, which will hold the confirmation hearings, and on Saturday he tweeted: "I fully understand where President @realDonaldTrump is coming from."
A confirmation vote before 3 November would be unusually quick.
The average period from nomination to confirmation is around 70 days.
But Trump has powerful incentive to push ahead, and not just to provide an electrifying jolt of enthusiasm among his anti-abortion and evangelical supporters.
The court could play a decisive role in legal wrangling over a contested election - as when the court decided in George W Bush's favor in the bitter election fight of 2 000.
Trump has already named scores of conservatives as federal judges, and Democrats are anxious about the lasting impact of a deep shift in balance at the Supreme Court.
"If he's allowed to put more conservatives in, this is going to be disastrous for the next 40 to 50 years," said Gloria Browne-Marshall, a civil rights attorney, on CNN.
Some Democrats have mused that if Biden is elected and Democrats control both houses of Congress, they might expand the court from nine seats to 11 - allowing the new president to appoint two more liberal justices.