Serena Williams, 33 this month, is the oldest player to be world No 1, but Billie Jean King believes she has years left at the top. King expects Williams to beat Caroline Wozniacki in today's US Open final and to win "many more" Grand Slams.
Victory would take Williams to 18 Grand Slam singles titles, level with Chris Evert and Martina Navratilova. Steffi Graf, who won 22, is the only player in the Open era who has won more.
"Serena will end up winning more than 18," King said. "After that the next goal will be Steffi Graf on 22. I can imagine her doing that. I told her years ago that I was only on 12. I said: 'If you don't do a lot better than 12 I'll be upset, because you could be the best ever'."
King, aged 70, regards Williams as the best of all time. She believes that only health or fitness problems, or lack of motivation, will stop her overhauling Graf's total.
"The players are getting older when they are winning," King said. "You need to be filled out and your body more mature because of the power. It's down to if she wants to pay the price. I played until 40. She hasn't had major operations. She is a phenomenal athlete. She gets a lot of free points on her serve."
As captain of the United States' Fed Cup team 1995 to 2003, King and her assistant, Zina Garrison, had a major influence on Williams. "Every time Serena played when Zina and I were coaches she won the next tournament," King said.
"We taught Serena sliced backhand. The first time she won here at the US Open [in 1999] was just after she had played Fed Cup for the first time. She won the final on two sliced backhands in the tie-break at the end. I'll never forget Zina running up to me, crying and saying: 'She hit those slices. I can't believe it'."
Billie Jean King will join Sir Elton John to stage Mylan Smash Hits at the Statoil Masters Tennis at the Royal Albert Hall on 7 December. The charity event will feature John McEnroe, Tim Henman and Kim Clijsters.
Tickets: http://www.statoilmasterstennis.com; 0207 070 4404