The Democrat told the crowd at the University of Miami: "A guy who ended up becoming our national leader said, 'I can grab a woman anywhere and she likes it.'
"They asked me if I'd like to debate this gentleman, and I said no. I said, if we were in high school, I'd take him behind the gym and beat the hell out of him.'"
He said any man who disrespected women was "usually the fattest, ugliest SOB (son of a b***h) in the room."
It's not the first time the pair have speculated about who would win if they came to blows.
While campaigning for Hillary Clinton in Pennsylvania in October 2016, Biden told the crowd: "I wish we were in high school — I could take him behind the gym. That's what I wish."
Trump, 71, returned serve by telling a Florida crowd he'd "love" to be given a shot at the former vice president. "I'd love that. I'd love that," he said. "Mr Tough Guy. You know, he's Mr Tough Guy."
Biden, 75, has kept open the possibility of a 2020 presidential bid and is set to play a major role campaigning for Democrats in this year's midterm elections.
Trump dismissed the prospect of a President Biden recently at the annual Gridiron Dinner with Washington journalists, calling him "Sleepy Joe" and saying he could "kick his ass".
And this has been going on for years. In 2016, the then Republican candidate attacked Biden on Twitter in 2016, calling him, "our not very bright Vice President."
And Biden told the audience at an Ohio Clinton rally he was "so sick and tired of hearing people like Trump and the Chamber of Commerce, the National Chamber, talking about we get paid too much."
As the crowd cheered, he added" "Give me a break. Give me a break. This is a guy born with a silver spoon in his mouth that now he's choking on because his foot is in his mouth along with a spoon."
Ironically, Trump also boasted about his shrewd use of social media on Twitter on Thursday morning.
