President Joe Biden has admitted it was a “mistake” to say he wanted to put a “bullseye” on his Republican election rival Donald Trump ahead of an assassination attempt on the former president.

In an interview with NBC News, Mr Biden argued that the rhetoric from his opponent was more incendiary while warning that Mr Trump remained a threat to democratic institutions.

The interview marked the resumption of campaigning for the Democrats following Saturday’s assassination attempt, which came after Mr Biden told donors in a private call that he was “done” talking about his poor debate performance and that it was “time to put Trump in the bullseye”, saying Mr Trump has received far too little scrutiny on his stances, rhetoric and lack of campaigning.

While he acknowledged his “mistake”, Mr Biden told NBC he is “not the guy who said I wanted to be a dictator on day one” and that he wanted the focus to be on what Mr Trump was saying, accusing his rival of engaging in that kind of rhetoric, referring to the former president’s past comments about a “bloodbath” if he loses in November.

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President Joe Biden walks to board Marine One on Monday (Jacquelyn Martin/AP)

“Look, how do you talk about the threat to democracy, which is real, when a president says things like he says?” Mr Biden said. “Do you just not say anything because it may incite somebody?”

The NBC interview, scheduled before the attempt on Mr Trump’s life at a rally in Pennsylvania, had been part of Mr Biden’s broader strategy to prove his fitness for office after his disastrous debate performance on June 27.

In the immediate aftermath of the assassination attempt, the Biden campaign pulled advertising off the air while the White House scrapped the president’s planned visit on Monday to the Lyndon B Johnson Library, where he had been due to deliver remarks on civil rights.

Mr Biden spoke privately to Mr Trump after the assassination attempt, a call that the president described in the NBC interview as “very cordial”.

“I told him how concerned I was and wanted to make sure I knew how he was actually doing,” Mr Biden said. “He sounded good. He said he was fine and he thanked me for calling.

“I told him it was literally in the prayers of Jill and me, and I hoped his whole family was weathering this.”

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Republican Ohio Senate candidate JD Vance speaks as Republican presidential candidate former president Donald Trump listens at a campaign rally (AP)

Mr Biden’s renewed campaigning comes as Democrats have been at an impasse over whether the incumbent president should continue in the race even as he was defiant that he would stay in.

Mr Biden has made it clear in no uncertain terms that he remains in the race and aides have been operating as such.

Asked by NBC interviewer Lester Holt if he has weathered the worst of it from his own party, Mr Biden responded that 14 million Democratic voters selected him through the primaries and added: “I listen to them.”

He said his mental acuity is “pretty damn good”, but said the question of his age was “legitimate” to ask.

The president and his campaign also criticised Mr Trump, who was confirmed as the Republican nominee for president, nominating freshman Senator JD Vance of Ohio as his running mate.

“He’s a clone of Trump on the issues,” Mr Biden told reporters as he headed to Nevada for a series of speeches and campaign events. “I don’t see any difference.”

The president’s staff issued a statement saying Mr Vance had been chosen because he would “bend over backwards to enable Trump and his extreme MAGA agenda”.