With ballots still being counted in several states, Mr Trump on Wednesday made an appearance at the White House and falsely declared victory against Democratic challenger Joe Biden.
Mr Trump railed against voting by mail during the election campaign, saying without providing evidence that it led to fraud, which is rare in US elections. Sticking to that theme, Mr Trump said: "This is a major fraud on our nation. We want the law to be used in a proper manner. So we'll be going to the US Supreme Court. We want all voting to stop."
Mr Trump did not provide any evidence to back up his claim of fraud or detail what litigation he would pursue at the Supreme Court. Later in the day, his campaign filed to intervene in a case already pending at the Supreme Court seeking to block late-arriving mail-in ballots in Pennsylvania.
The Trump campaign and other Republicans have also filed various complaints in other states, including an attempt to stop votes being counted in Michigan.
As of midday on Thursday, the election still hangs in the balance. A handful of closely contested states could decide the outcome in the coming hours or days, as a large number of mail-in ballots cast amid the coronavirus pandemic appears to have drawn out the process.
However, legal experts said that while there could be objections to particular ballots or voting and counting procedures, it was unclear if such disputes would determine the final outcome.
Ned Foley, an election law expert at Ohio State University, said the current election does not have the ingredients that would create a situation like in the 2000 presidential race, when the Supreme Court ended a recount in George W Bush's favour against Democrat Al Gore.
"It’s extremely early on but at the moment it doesn’t seem apparent how this would end up where the US Supreme Court would be decisive," Mr Foley said.
Both Republicans and Democrats have amassed armies of lawyers ready to go to the mat in a close race. Mr Biden's team includes Marc Elias, a top election attorney at the firm Perkins Coie, and former Solicitors General Donald Verrilli and Walter Dellinger. Mr Trump's lawyers include Matt Morgan, the president's campaign general counsel, Supreme Court litigator William Consovoy, and Justin Clark, senior counsel to the campaign.
Mr Trump's attorney Jenna Ellis on Wednesday defended his bid to challenge the vote count and evaluate his legal options. "If we have to go through these legal challenges, that's not unprecedented," Ms Ellis told Fox Business Network in an interview. "He wants to make sure that the election is not stolen."
The case closest to being resolved by the Supreme Court is the Pennsylvania dispute in which Republicans are challenging a September ruling by Pennsylvania's top court allowing mail-in ballots that were postmarked by Election Day and received up to three days later to be counted.
The Supreme Court previously declined to fast-track an appeal by Republicans. But three conservative justices left open the possibility of taking up the case again after Election Day.
Even if the court were to take up the case and rule for Republicans, it may not determine the final vote in Pennsylvania, as the case only concerns mail-in ballots received after 3 November.
David Boies, who represented Mr Gore in 2000, said it is unlikely that the Trump campaign would succeed in a possible third effort to block the extended deadline.
"I think that it’s more posturing and hope than anything else," Mr Boies said, adding that the Pennsylvania outcome could even become irrelevant, depending on the result in Michigan and Wisconsin.
In a separate Pennsylvania case filed in federal court in Philadelphia, Republicans have accused officials in suburban Montgomery County of illegally counting mail-in ballots early and also giving voters who submitted defective ballots a chance to re-vote.
If Mr Biden secures 270 electoral votes without needing Pennsylvania, the likelihood of a legal fight in that state diminishes in any case, legal experts said.
And any challenge would also need to make its way through the usual court hierarchy.
"I think the Court would summarily turn away any effort by the President or his campaign to short-circuit the ordinary legal process," said Steve Vladeck, a professor at the University of Texas at Austin School of Law.
"Even Bush v. Gore went through the Florida state courts first."