US Republican presidential candidate Donald Trump ratcheted up his attacks on Vice President Kamala Harris at a rally near Miami on Tuesday, trying to discredit her as the Democratic Party tussles over whether President Joe Biden should remain in the race.

Speaking at his golf club in Doral, Florida, Trump stayed mum on his planned choice for his running mate, although he name-checked Florida Senator Marco Rubio, who was in attendance and is a leading contender for the role, multiple times.

Taking the stage nearly an hour late on a hot night, Trump blasted 81-year-old Biden about his shaky debate performance on June 27, which prompted worries about his mental fitness for office and calls from some Democrats for him to step aside.

Biden said he had a bad night and that he will not drop out of the presidential race.

In a mocking tone, Trump called Biden's choice of Harris, the first black and woman vice president, his "one brilliant decision," implying that she was so inept that Democrats were reluctant to replace Biden with her on the top of the ticket.

"It was an insurance policy, maybe the best insurance policy I've ever seen," Trump said. "If Joe had picked someone even halfway competent, they would have bounced him from office years ago."

Trump also accused Harris, a former US senator and California attorney general, of working with other Democrats on covering up Biden's alleged mental acuity issues. Biden has said he has the stamina and mental sharpness to do the job.

Trump also said Harris's record would hurt her electoral chances, the latest sign that he and his Republican allies are starting an all-out assault on Harris to prepare for Biden possibly ending his re-election bid.

"I don't think Kamala Harris's California socialism is going to go down well with the people of Doral, the people of Miami or the people of Florida. Because in Florida we don't like socialism, we want our freedom," Trump said.

The Biden campaign accused Trump of lying about Harris, and said he was trying to deflect from growing attention paid to Project 2025, a conservative group's controversial plans to remake the presidency should Trump win the November 5 election.

"Trump's bizarre and unhinged rants won’t stop Vice President Harris from defending the Biden-Harris record and prosecuting the case against Donald Trump’s extreme record," Biden campaign spokesperson Ammar Moussa said.

Golf challenge

Seemingly joking, Trump challenged Biden to another debate, saying that it should be "man to man, no moderators, no holds barred," and to 18 holes of golf - a reference to their argument during the debate over who was better at the sport.

"It will be among the most watched sporting events in history, maybe bigger than the Ryder Cup or even the Masters," Trump said, adding that he would give Biden 20 strokes.

The Biden campaign dismissed Trump's golf challenge as a stunt. They said Biden, who gave a speech at the Nato summit in Washington on Tuesday vowing to defend Ukraine against Russian President Vladimir Putin's invasion, was too busy governing.

"We'd challenge Donald Trump to stand up to Putin, but he bent the knee to him," said campaign spokesperson James Singer.

Trump praised what he called an "incredible decision" by the Supreme Court last week determining that he could not be prosecuted for actions within his constitutional powers as president, a landmark decision that could have broad implications for the criminal charges he is facing.

Biden called the ruling "a dangerous precedent" because the power of the presidency will no longer be constrained by law.

Calling the prosecutions against him politically motivated, Trump said he had "great respect for the courage" shown by the conservative justices who ruled in his favour. "They have great intellect and great insight, incredible insight," he said.

On Saturday, the former president will travel to Pennsylvania for an afternoon rally in Butler, just outside of Pittsburgh, where the focus will be on any hint - or announcement - about whom he will pick as vice president.

Rubio, Ohio Senator JD Vance and North Dakota Governor Doug Burgum are seen as the leading contenders to join Trump on the Republican ticket. Trump said this week that he would likely make a decision "a little before" or during the Republican Party's national convention, which kicks off on July 15.

Calling out Rubio by name, Trump pointed out the large number of journalists in attendance. "I think they probably think I'm going to be announcing that Marco is going to be vice president, because that's a lot of press," he said.