And he thought Street's shoes were big.
The 21-year-old right-hander won't be counted on to take over for Rivera for at least a year or two -- if ever, that is -- but the mere idea of it seems so unreal to Cox, he seems uncomfortable even talking about it.
"You like to dream about it, but you have to let it play the course," Cox said. "To be a big-league pitcher would be great, but to be a closer would be the icing on the cake. I just want to get a chance, whether it's as a closer or not.
"Mariano has some gas left in his tank," he added. "He's not going anywhere for a while."
Very true, especially when you consider that Rivera posted what may have been his best season in 2005. But Cox isn't spending any time worrying about what his future may bring; he's more concerned about the present.
Cox is in the Yankees' big-league camp this spring, less than one year after being selected in the second round of the 2005 First-Year Player Draft. He wears No. 79, and his locker is in the middle of the room with the rest of the guys wearing numbers that resemble offensive linemen and tight ends.
Not that he's complaining. After all, he's had a chance to coexist with some of the biggest stars in the game, getting a first-hand look at what life in the Majors is all about.
"Everybody has been real nice," Cox said. "Coming in, I expected it to be more serious. It's serious -- don't get me wrong -- but it's more laid back than I thought it would be."
Veterans such as Al Leiter and Ron Villone have taken some time to speak with Cox and some of the young pitchers, talking about their approach to the game, how to handle life in the Minors and what it takes to get through the long baseball season.
"To be able to learn, to pick at some of the veterans' brains, it's pretty cool," Cox said. "Some of these guys have been around a long time, so it's amazing to listen to what they have to say, some of the ins and outs of the game."
Cox's sidearm delivery has prompted comparisons to Jeff Nelson, who was an integral part of the Yankees' bullpen during their championship run of the late-1990s. Manager Joe Torre has been impressed with what he has seen from Cox, and he believes that he has what it takes to succeed once he arrives in the Majors.
"He's got the ability, being a sidearmer, to get left-handers out. That's a feather in your cap," Torre said. "He seems to have good makeup. He's very aggressive, he's young and he's learning."
The Yankees liked Cox so much before last June's draft, they passed on the opportunity to take Craig Hansen, who had been the closer at St. John's University. New York selected high school shortstop C.J. Henry instead, believing that they would be able to land Cox with their second pick.
"They saw me pitch in big games, in pressure situations at Texas," Cox said. "That's probably one of the reasons I got invited here after not playing much Minor League ball. My maturity level may be a little higher than some of the younger guys, and that comes from my experience at Texas."
Cox also learned about high expectations and overwhelming media attention while pitching for the Longhorns, so being a part of Yankees camp isn't quite as foreign for him as it may be to some of his fellow Minor Leaguers.
"I came from a place where there are nothing but high expectations; if we didn't win the College World Series every year, we got criticized," Cox said. "We had more media at Texas than most other college players had, so overall it was kind of a blessing that I went to Texas. It helped prepare me to get drafted by the Yankees."
Cox pitched for Class A Tampa in the Florida State League after being drafted last season, going 1-2 with a 2.60 ERA in 16 games. He will likely start this season with Double-A Trenton, though many believe he could find himself in the Bronx before the year is finished.
"With the Yankees, you have to be patient," Cox said. "Not too many doors are swinging open."
But when one does, Cox plans to do what he does best: close it.