Mr Clark says he and Mr Dean met in early September when the vice-president's slot "was sort of discussed ... and dangled before I made the decision to run". He said he told the former Vermont governor he was "really not interested in even talking about it".
He added he was "absolutely not" interested in being Mr Dean's running mate. "I don't see that in the cards," he said.
The Dean campaign denies that the offer was ever made. Joe Trippi, Mr Dean's campaign manager, said Mr Clark's account of the meeting was "interesting, but not the meeting I was in. That never came up ... In the period he was talking about, we were still an asterisk in most of the polls, so to be talking to anybody about being vice-president doesn't make a whole lot of sense."
The Clark campaign replied that the offer came "in a one-on-one meeting that Trippi did not attend".
Mr Dean also denied yesterday that he had asked Mr Clark to join him. "No I did not," he told Reuters news agency. "Wes is a good guy. He'd be a good running mate, but I think it would be highly foolish of me to offer anybody the running mate spot."
So far Mr Clark has been the only Democratic presidential contender who has not singled out Mr Dean for attack since the former governor became the party's frontrunner. With the exception of a comment last month that while he was getting wounded in Vietnam, Mr Dean (who had failed his physical with a back problem) was skiing in Aspen, Mr Clark has tried to stay above the internal schisms.
Some in both camps believe the two men would complement each other well. Mr Dean, who has no hands-on foreign policy experience, has used his opposition to the war in Iraq to fuel his presidential bid. Mr Clark, a former Nato chief who directed the 1999 bombing in the Kosovo war, has impeccable military credentials and remains sceptical about the Iraq war.
Mr Dean, who received the endorsement of the former vice-president Al Gore, has come to represent the insurgent populist wing of the party. Mr Clark, who is believed to enjoy the private endorsement of both former president Bill Clinton and his senator wife, Hillary, is backed by the party establishment.