"The initial report is not good," Girardi said.
Nady, a free agent-to-be who was acquired in a trade with the Pirates last July, was visibly emotional as he discussed the injury in front of his locker in the visitors' clubhouse at Tropicana Field.
"I'll try to be as optimistic as I can and be back out there as quick as I can," Nady said. "It hasn't necessarily sunk in. It's frustrating, obviously."
Nady suffered the injury in the seventh inning of Tuesday's 7-2 win over the Rays. He felt a sharp pain making a throw after a Carlos Pena single.
Nady, 30, has experience with elbow injuries. He underwent Tommy John elbow reconstruction surgery as a Minor Leaguer in 2001. That year, his surgery was performed after the season, and he spent the offseason in rehabilitation.
Nady said he was ready to hit again by Spring Training 2002.
"I've obviously been down this road before," Nady said. "I was hoping not to do it again. I'll just see what [Ahmad] thinks [Thursday]. I would prefer not to go down the surgery route. Hopefully, tomorrow there will be some decent news and we'll go from there."
Girardi said that Nick Swisher will step in to play right field in Nady's absence.
No decision had been made as of Wednesday afternoon regarding the player expected to replace Nady on the roster.
"That's something we'll discuss," Girardi said. "We'll talk about it with [general manager Brian Cashman] probably on the way to the airport today, talk about it with the coaches and decide what we're going to do."
Girardi did say that he wasn't inclined to add an outfielder with Swisher, Johnny Damon, Melky Cabrera and Brett Gardner already in the fold. An infielder or a pitcher would be possibilities, Girardi said.
"We don't have as much depth, is the first thing you would look at," Girardi said. "We talked about that we had five outfielders to play [three] spots. Now, we've got four to play three, depending on what we do. We just don't have the depth that we used to have. It's something we might have to deal with for a while."
Nady was batting .286 with two RBIs and four doubles at the time of the injury. In 59 games after the trade, he batted .268 with 12 home runs and 40 RBIs with the Yankees.
"He's been a clutch player the 2 1/2 months that we've had him," Girardi said. "He's been a big part of our lineup. He's a great human as well. It's frustrating, but we'll have to wait."