Karen Hughes made the debate proposal in Austin, Texas, as Gore accepted his Democratic presidential nomination in the finale of his party's national convention.
Gore has made debate challenges part of his strategy, first against Democratic rival Bill Bradley, then against Bush after the presidential nominations were clinched. The Gore campaign has accused Bush of ducking.
"We hope the Bush campaign really wants to debate," Chris Lehane, Gore's spokesman, said Thursday. "We've been ready to debate for months. Their newfound readiness to debate shows this campaign is tightening."
The Texas governor said after his nomination two weeks ago that he wanted debates but hadn't settled on a number or a format. "I think it's going to be healthy for our country to have debates," he said.
There have been two or three presidential debates in each campaign since 1976, one between vice presidential nominees since 1980.
The Bush proposal is to add a second debate between his running mate, former Secretary of Defense Dick Cheney, and Gore's, Sen. Joseph Lieberman of Connecticut.
In the interview, Bush avoided answering the question of whether Pat Buchanan and Ralph Nader, running on the Reform and Green party tickets, should be part of the debates. They're both demanding it; the rules set by a private panel that has sponsored prior debates would keep them out.
Gore hasn't taken a position on that, either. Both the major party nominees have said they think the voters would like to see them face to face. "I do want an opportunity to debate Al Gore one on one," Bush said in the AP interview.
The Commission on Presidential Debates has proposed three between the presidential candidates, on Oct. 3 in Boston, Oct. 11 in Winston-Salem, N.C., and Oct. 17 in St. Louis. The commission plan is for a vice presidential debate in Danville, Ky., on Oct. 5.
The debate commission set a threshold of 15 percent in the national public opinion polls for inclusion in the debates, and neither Buchanan nor Nader is close. Ross Perot got into the 1992 campaign debates and sued, unsuccessfully, when he was excluded in 1996.
The commission has sponsored presidential campaign debates since 1988, proposing times, places and formats, always subject to negotiations by the candidates.