Washington DC - President Joe Biden sought to project calm Sunday after the attempted assassination of Donald Trump, saying in a rare Oval Office address that it was time to lower the temperature of politics in the US.

"It's time to cool it down. We all have a responsibility to do that," Biden said in a televised address following the attack, in which Trump was injured in the ear and a bystander was killed by gunfire.

As the country reeled from images of a bloodied Trump waving his fist after the gunman opened fire at a campaign rally in Pennsylvania, Biden added that US politics "must never be a literal battlefield, God forbid a killing field."

The 20-year-old shooter, Thomas Matthew Crooks, was killed by Secret Service agents amid scenes of chaos. Authorities say his motive remains unclear.

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The FBI said it was investigating the attack as a potential act of domestic terrorism and studying Crooks's phone to discover any "ideologies" he may have had.

Biden, giving just the third Oval Office address of his presidency, also mentioned the January 6, 2021, Capitol attack by pro-Trump supporters as proof that the situation is getting out of hand.

"We can't allow this violence to be normalized," the 81-year-old said, adding that the November 5 election would be a "time of testing" for the US.

The short but forceful speech went without any major hitches – bar Biden twice referring to the ballot box as a "battle box". Democrats are closely watching the president following a disastrous debate performance renewed concerns about his age and ability to govern.

The attempt on Trump's life has opened a dark new chapter in an already supercharged election rematch with Biden, who beat him in 2020.

Despite their bitter enmity, both Biden and Trump called for calm after the most serious attack on a US president or ex-president in more than four decades.

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Trump said it was "more important than ever that we stand United" and added that Americans should not allow "evil to win," in a post on social media, before telling the New York Post that he was rewriting his speech for Monday's Republican National Convention to focus on unity.

Biden, who is behind in most polls, will pick up his campaign with a visit to the battleground state of Nevada later this week.

"That's how democracy should work," he said in the Oval Office address. "We debate and disagree."