He is determined to be known as the greatest, period.
60 Minutes II Correspondent Charlie Rose first described his dream last January. At Wimbleton this past summer he won his 13th Grand Slam title, the most in men's tennis history.
And if he wins this week at the U.S. Open, it will be his 14th Grand Slam. You have to wonder, how has he done it?
What is it that makes Sampras so special? Well for starters, just try returning his serve.
Sampras dominates his sport as no other athlete does. He can rifle a serve at 130 mph. He is rich and good-looking. But he has never set the public's heart on fire.
"I'm more of a '50s kind of old-fashioned type of guy," says Sampras, who nevertheless disagrees that he is boring, as so many of his critics claim. "I'm not a flamboyant, talkative, I-want-to-pat-myself-on-the-back type of guy."
Throughout his career, Sampras has been criticized for the things he is not. He is not obsessed like Bjorn Borg, not flashy like Jimmy Connors and not controversial like John McEnroe.
Sampras says that unlike those players, he doesn't feel that his happiness depends on being the top-ranked player. It matters to him, but not that much, he says. "But to stay No. 1 it - it's got to be your life, it really does."
If Sampras does not excite tennis fans by himself, his rivalry with Andre Agassi does. The Sampras-Agassi rivalry has revitalized men's tennis. Part of that is the drama of Agassi's comeback: Last year he rose from 141st to the top of the rankings.
Whenever they play, Sampras raises the level of his game.
"I like what I feel when I'm on the court with Pete," Agassi says. "I like what I feel when I'm in anticipation of playing him. Knowing that you're going to have to dig down to a place where, quite potentially, you've never dug before."
"I play other guys, and I know, 'OK, if I just do this, (if) I focus this hard, if I'm this intense, if I do this, I'm going to win.' And with Pete, you go, 'I got to do that, and then I don't know,'" he says.
In the summer of 1999 Sampras was poised for another epic battle with Agassi, in the U.S. Open. But after hurting his back, Sampras had to withdraw and watch as Agassi won the U.S. Open. Now with his back healed, he's again at the top of his game.
But they leave their rivalry on the court. "There's never been tension with Andre and (me) off the court," Sampras says. "We like each other."
Sampras and Agassi have been competing against each other since they were kids. Sampras grew up in Southern California and first picked up a racquet when he was 7.
He led a single-minded life. "It was really - just an amazing upbringing," he says. "Playing tennis every dy, three hours a day. And (I) didn't have any girlfriends, no social life. Didn't go to the prom. I just - it was tennis, tennis, tennis."
Sampras says that his parents didn't pressure him, but encouraged him.
Because his parents are superstitious, they don't attend his matches. In fact, when Sampras won his first U.S. Open nine years ago, they went to a shopping mall so they wouldn't see him play. They have seen him play once - in the 1992 U.S. Open final, which he lost to Stefan Edberg.
His parents also shun the spotlight. "They are very private people," Pete Sampras explains. "They are like me, in a way." Even so, Sampras wishes that his parents would watch his matches in person.
Despite his reputation, Sampras says that he enjoys the spotlight. "It's fun to play in front of tons of people," he says. "It's a great feeling. It's funny, when I break my string and I change rackets, I take my time. I'm not freaked out. I don't have to rush and get a new racket. This is my stage."
According to McEnroe, Sampras has that ability to focus that distinguishes all great champions.
"In some ways, that's why he's called boring," McEnroe says. "Because he's so into it that he's totally put aside virtually anything else, which in some ways is something that I wish I could have done a better job at. The bad news is that you're thought of in this - in (a) negative light. The good news is that you rack up majors."
In 10 or 15 years, McEnroe says, Sampras' greatness will be truly recognized. The problem, says McEnroe, is that Sampras is so good that he makes the game look easy.
McEnroe thinks that Sampras could even have beaten Rod Laver, the Australian player thought by many to have been the greatest player ever.
Sampras idolizes Laver and has since he was a boy. He was taught to revere the Australian by Dr. Pete Fischer, a pediatrician who was Sampras' first coach. Fischer told him that he could be better than Laver, Sampras says.
Fischer was a demanding taskmaster, and after a decade together, coach and player split.
And Sampras did not know that his coach had a secret life. In 1997, Fischer pleaded guilty to molesting several of his young patients. He is now serving a six-year sentence in a California prison.
Sampras, who says that he had no idea his former coach was a child molester, has not not abandoned Fischer. "I'm supporting him as a friend," he says. "I've known Pete since I was 7 years old. And when he gets out of prison, I'm going to be his friend."
Indeed Sampras is loyal. After leaving Fischer, he hooked up with coach Tim Gullickson, who was by his side when Sampras began to dominate the sport with cool precision. Then, at the Australian Open in 1995, Sampras learned that Gullickson, by now a close friend, was dying of brain cancer.
He broke down. Many were surprised that a man known for his precise demeanor could be so emotonal.
Off the court, Sampras is polite with fans, but hardly outgoing. He doesn't fit the image of a superstar.
But Sampras says that he is not an automaton.
"I'm a very emotional person; I really am," he says. "I internalize a lot of my feelings. I'm not a robot. I'm not playing and winning tournaments and winning matches because I'm a robot. That was a lot of hard work and passion that I have going into this game."
Sampras, who wants to play until he is 40, is now focusing on the Australian Open. Tuesday, Jan. 25, he took a step closer to breaking the men's Grand Slam record, winning his quarterfinal match.
His dream finale: He is playing in the finals against Agassi, with Laver in the audience. And his parents are there. "Maybe I'll get them to come," he says. "That's a fantasy."
September 2000 Update: After that interview, Pete Sampras finally realized that dream.
In July, he broke the men's Grand Slam record by winning Wimbledon.
Immediately afterward, Sampras climbed into the stands to embrace his parents, who were there to watch their son make tennis history.