In what appeared to be a sexual innuendo, the president said: “For 47 years, Joe Biden took the donations of blue collar workers, gave them hugs and even kisses ...”
As members of the audience laughed and applauded, Mr Trump paused, raised his eyebrows and smirked.
Mr Biden has been praised for his empathetic rapport with people but has also been criticised by some as being overly tactile.
In March this year he was accused of having committed a sexual assault against a former staffer, Tara Reade, in 1993 when he was still a senator. He has repeatedly denied the accusation.
Mr Trump meanwhile has been accused of a series of sexual assaults, harassment and even rape by more than 20 women. He too has denied any wrongdoing.
Shortly before the 2016 election a video outtake from the TV show Access Hollywood was leaked showing Mr Trump boasting about sexually assaulting women.
The resulting row posed a serious threat to his election chances and apparently prompted the leak of hacked Democratic emails suggesting the party had conspired to help Hillary Clinton win the nomination over Bernie Sanders.
Mr Trump insisted that his boasts were nothing more than “locker talk”.
The president’s joke at Mr Biden’s expense came during his acceptance speech at the Republican National Convention, controversially held at the publicly-funded White House in an apparent breach of ethical norms.
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