Mr Biden is currently leading the opinion polls in the race to win the Democratic Party’s nomination to fight against Mr Trump for the keys to the White House.
The American politician made the claim to leading donors of his campaign.
At a private fundraiser on Saturday night in Columbia, South Carolina, Mr Biden spoke in the private home of state Senator Dick Harpootlian, an attorney and former South Carolina Democratic Party chairman.
He said “at least 14 world leaders” have called him during Mr Trump’s presidency to express their concerns.
The former vice president asked him for reassurance the UK and the US “still have a special relationship”.
Mr Biden said the US under the Republican firebrand “is about to squander alliances” built over generations.
Mr Biden has nicknamed Mr Trump a “clown” after a supporter at the fundraiser asked if he would return the President’s insults.
He told around two dozen supporters at a private fundraiser he didn’t want to get down in the mud with the billionaire.
Mr Biden said: “There are so many nicknames I’m inclined to give this guy.
“You can just start with clown.”
Since launching his campaign, Mr Trump has attacked Mr Biden, calling him “sleepy Joe”.
The nickname refers both to the presidential hopeful’s age and the length of time it took him to begin his campaign.
Mr Biden said although he will respond to Trump if directly attacked, he believed it was part of the President’s strategy to keep dialogue away from the real issues.
“On every single issue and on every demeaning thing he says about other people, I have no problem responding directly,” Mr Biden said.
“What I’m not going to do is get into what he wants me to do.
“He wants this to be a mud wrestling match.”