"They're a joke. They're a laughingstock," Trump told reporters when asked whether he would agree to debate them during the 2020 nominating contest.
The Republican, who for years now has steadfastly refused to release his tax returns despite a long history of presidential candidates detailing their finances, also said he planned at some point to put out an "extremely complete" statement on his finances.
He said the financial statement would make clear he does not need whatever revenue was produced when U.S. military personnel stayed at his Turnberry golf resort in Scotland while on refueling stops.
"I'm going to give out my financial condition, and you'll be extremely shocked that the numbers are many, many times what you think. I don't need to have somebody take a room overnight at a hotel," he said.
Three Republicans - former U.S. Representative Mark Sanford of South Carolina, former Massachusetts Governor William Weld and former U.S. Representative Joe Walsh of Illinois - are mounting long-shot campaigns to deny Trump the 2020 Republican presidential nomination.
In a tweet, Trump recalled Sanford's controversial disappearance in 2009 when, while governor of South Carolina, he met his Argentine mistress under the cover story of having gone hiking on the Appalachian Trail, only to be found out later.
"The Three Stooges, all badly failed candidates, will give it a go!" the president said. The Three Stooges were a vaudeville and slapstick comedy team whose antics regularly appeared on television.
Trump also denied having anything to do with the cancellation of Republican primary contests in four states, meaning he will face no opposition there. Canceling their primaries were Nevada and South Carolina, which are critical early voting states, as well as Kansas and Arizona.
"The four states that canceled it don't want to waste their money. If there was a race, they would certainly want to do that," Trump told reporters on the South Lawn of the White House.