"Excuse me, but you haven't won the match yet," whistled back the bespectacled Texan, his eyeballs staring down the fairway. "There are still four games out on the course and if we win them all then it will be a tie." That proved one Rider Cup the Americans did win as Kite's prediction proved spot on.
Seven years later, Kite is installed as the captain of the United States team and the countdown to next year's contest at Valderrama in southern Spain has begun. After handing in a 69, "which could have been a 63 or 64", after his third round at the Open yesterday to stand on 212, one under par, his thoughts turned to the Ryder Cup - and not just the 1997 version. Looking to 2001, he was critical of the decision to stage the match at the Belfry for a fourth time.
"I was surprised by that. There are so many great venues around Europe that would love to have a shot at hosting the Ryder Cup; why should it keep going back to the same course? I thought it was a bit short-sighted of the PGA.
"We have done an excellent job in the States, moving the Ryder Cup around so that people all over the country get a chance to see it. It's difficult for us to take it to the West Coast because we want the people over in Britain and Europe to see it on television at a reasonable hour. The same approach ought to be taken here. Certainly, moving it to Spain was an excellent step."
Kite's participation on the final two days at Royal Lytham had been in doubt after an opening 77, but on Friday a typically battling 66, in which he played the final 15 holes in six under par, brought him in on the cut-off mark of 143, one over. His superb start yesterday, four under after seven, meant he had covered 22 holes in 10 under - not quite enough to threaten the upper echelons of the leaderboard but still the most impressive stretch of scoring in the Championship thus far.
"After that start I expected to do a lot better than 69," Kite said of yesterday's round. "I had a lot more birdie chances but after the seventh nothing went in. I also had a couple of bogeys - at the eighth, where I drove into the right rough, and at 15, where I three-putted. At one under, I am too far back to get into contention, but it's great to see so many Americans doing well here, especially Jack Nicklaus.
"I'm enjoying being captain and I'm going to have a lot of fun over the next 14 months. I'm planning to come over to see Valderrama in October. I'm playing in a Pro-Am in Madrid and will go down there afterwards. We'll try to get something going there in conjunction with next year's Open and get some of the players who look like making the team to go and see the course."
Even though his 47th birthday is only a few months away, Kite's fighting spirit burns as fiercely as ever. He is golf's second-highest prize money winner ($9.6m at the last count) and has no intention of easing up, especially with the Senior Tour gravy train just over the horizon. For years he came close, but not close enough, at major after major before winning the US Open at Pebble Beach in 1992, a breakthrough which set the seal on his Ryder Cup record and 18 other tour titles.
Not only is Kite a gritty competitor, he is an eternal optimist. "My form's very good at the moment, and I'm definitely ready to win again soon," he said. And who would argue against him? Certainly not Steve Rider.