Trump, 78, was holding a campaign rally in Butler, Pennsylvania - a key state in the November 5 election - when shots rang out, hitting his right ear and leaving his face streaked with blood. Trump, as a former president, has lifetime protection by the US Secret Service.
Both Trump and Biden on Sunday sought calm and unity.
Trump is due to accept his party's formal nomination at the Republican National Convention with a speech on Thursday. "This is a chance to bring the whole country, even the whole world, together," Trump told the Washington Examiner.
Biden delivered a televised address from the Oval Office in the White House on Sunday, saying the political rhetoric in the US has gotten very heated and it’s time to cool it down.
The FBI identified Thomas Matthew Crooks of Pennsylvania as the suspect and said the shooting was being investigated as an attempted assassination.
FBI officials said on Sunday that the shooter acted alone and the agency had yet to identify an ideology linked to the suspect or any indications of mental health issues or found any threatening language on the suspect's social media accounts.