Robert F. Kennedy Jr has apologised after his son shared a video of Donald Trump apparently trying to woo the high-profile independent presidential candidate.

In the accidentally leaked clip, Trump can be heard discussing Saturday’s attempted assassination, vaccines and babies – and making promises to Kennedy.

“I would love you to do something – and I think it would be so good for you and so big for you,” Trump says, apparently referring to the 2024 election race.

The footage was quickly deleted after being posted by Kennedy’s son Bobby Kennedy III on Tuesday (US time) – but not before it had been shared by multiple other accounts.

It shows Kennedy, a long-shot independent candidate in November’s election, on speaker phone with Trump while standing in a room with an American flag.

Trump can be heard telling Kennedy, who is a noted anti-vaxxer: “I agree with you, man. Something’s wrong with that whole system, and it’s the doctors you find.”

“Remember I said, ‘I want to do small doses. Small doses’,” Trump adds, echoing comments about vaccinating children that he made in 2015.

“When you feed a baby, Bobby, a vaccination that is like 38 different vaccines, and it looks like it’s meant for a horse, not a, you know, 10-pound or 20-pound baby.”

Later in the conversation, Trump says: “We’re gonna win.”

“Yeah,” Kennedy replies.

“We’re way ahead of the guy,” Trump adds, referring to Democrat President and candidate Joe Biden.

Trump can also be heard talking about the attempt on his life at a rally in Pennsylvania, saying the bullet that grazed his ear felt “like the world’s largest mosquito”.

“It was a, what do they call that, an AR-15 or something — that’s a big gun,” he said.

At the Republican National Convention late on Monday, Trump made his first public appearance since the shooting, sporting a large white bandage covering his ear.

On Tuesday, Kennedy apologised for the leaked conversation.

“When President Trump called me I was taping with an in-house videographer,” he wrote.

“I should have ordered the videographer to stop recording immediately. I am mortified that this was posted.”

Kennedy is the nephew of assassinated US president John F. Kennedy. His famous name and loyal base means he has the potential to do better than any third-party presidential candidate since Ross Perot in the 1990s.

Last year he challenged Biden for the Democratic nomination before launching his independent bid, and has since argued that his relatively strong showing in some polls gives his candidacy heft.

Kennedy is a long shot to win Electoral College votes, much less the presidency, but his campaign events have drawn large crowds of supporters.

This week, after the Trump shooting, Biden directed the US Secret Service to protect Kennedy. Homeland Security Secretary Alejandro Mayorkas said Biden had directed the protection “both prior to and after the events of this past weekend”.

The Secret Service is legally required to protect major party presidential and vice-presidential candidates and their families 120 days out from a general election. Third-party candidates are treated on an as-needed basis.

Threats to political candidates are common in the US. Law enforcement officials have also said there has been an uptick in violent rhetoric since the weekend attack at the Trump rally.

Mayorkas said both Biden and Trump were “constantly the subject of threats”.

“We are in a heightened and very dynamic threat environment,” he said.

Trump became the official Republican presidential nominee on Monday after receiving the votes of enough delegates at the Republican National Convention. He was not seriously injured in the shooting over the weekend in Pennsylvania.

An independent review into the response to the attack is under way.

Mayorkas said Trump’s protection had been enhanced based on the “evolving nature of the threats to the former president” and his shift from presumptive nominee to nominee.

-with AAP