The opportunity would be extremely meaningful for the 43-year-old Rivera, who said that he will only take the field if he feels that his body will allow him to play the position well.
"This is not a game for me, guys," Rivera said. "I don't know how you guys take it, but this is not a game for me. This is something I respect. I requested it way back, a long time ago, and now my knee is back. So we'll see what happens."
Girardi plans to use Rivera on Thursday, in what will be his final Yankee Stadium appearance.
Girardi would not consider playing Rivera in center field against the Rays, because the contest has playoff implications for Tampa Bay. The Astros are out of the race, so all three games of the weekend series in Houston are in play.
"You know, I don't know how I would do it," Girardi said. "It's something I would have to talk to him about. In my mind, thinking that he's going to want to pitch, it's a situation where I might bring him in in the eighth to play the outfield and then have him close it out in the ninth if we have that opportunity."
General manager Brian Cashman said on Thursday that he has no issues with giving Rivera his appearance in the outfield.
"It's whatever Mo wants, as far as I'm concerned, in Houston," Cashman said. "If that's something of interest, I'm sure he and Joe will work it out."
Rivera has long held an affinity for shagging fly balls, which he has called his favorite part of the afternoon before games. He sustained a catastrophic knee injury doing so last May in Kansas City, but he vowed to continue because he loves it so much.
There are also stories about Rivera's defensive prowess as he came up through the Minors, and some Yankees coaches believed he was among the organization's best athletes in the outfield. But he has never had the chance to play the outfield in a spring or regular-season game because of the risk that the Yankees might lose his services as a pitcher.
"It would be something good, if my body permits it," Rivera said. "The one thing that I will tell you, if I can do it, I will do it."
Girardi said that the unorthodox center-field hill at Minute Maid Park would not make him hesitate to put Rivera out there.
"I'm going to try to warn him about it, and hopefully, we don't give any balls up on that hill," Girardi said.
Rivera's possible appearance in the outfield is reminiscent of the pair that Ron Guidry made for the Yankees in center field.
Manager Billy Martin allowed Guidry to play the ninth inning of a Sept. 29, 1979, game against the Blue Jays, and nearly four years later, on Aug. 18, 1983, Martin sent Guidry to center field in the resumption of the "Pine Tar Game" against the Royals, also playing Don Mattingly at second base in mock protest.
Rivera has made it clear that he views this with more responsibility than Martin did.
"If I cannot do it, I will not be making a fool of myself there," Rivera said. "I'm a professional. This is not a joke. This is serious, this is business."