''I don't want to play any games with you. It's a process that's requested to be private,'' Kerry told reporters. ''I've committed that it's a private process.''

Kerry has been one focus of speculation as Gore mulls his choices for the Democratic ticket. Kerry, 56, is a Vietnam veteran who is the top Democrat on the Senate Small Business Committee. He has been in the Senate since 1985.

Others prominently mentioned as possible Gore running mates are former Sen. George Mitchell of Maine, Florida Sen. Bob Graham and Iowa Sen. Tom Harkin. House Minority Leader Richard Gephardt of Missouri and California Gov. Gray Davis repeatedly have said they don't want the job.

Gore's Republican opponent, Texas Gov. George W. Bush, on Tuesday named former Defense Secretary Dick Cheney as his running mate.

Kerry shared a podium at the Senate news conference with Sen. John McCain, R-Ariz., the lead sponsor of an Internet privacy measure Kerry also supports. McCain, a failed GOP presidential candidate, was not as reluctant to discuss the running mate question.

''He'd be very helpful to Vice President Gore,'' McCain said. ''Senator Kerry has proved his service to this country and ability as an accomplished debater.

''So for that reason, I strongly recommend that Gore not select Senator Kerry,'' McCain joked.