Overbay, signed to a minor league deal, will be considered as a potential fill-in for Mark Teixeira.
Teixeira is not expected to return to the field until some time in May at the earliest because of a torn tendon sheath in his right wrist.
"He's going to get an opportunity to play some first," Yankees manager Joe Girardi said. "He's been a very good first baseman over the years and he's going to get an opportunity to show us what he can do. We want to see how he moves around, see how he swings the bat, try to make a quick judgment. We don't have a lot of time. We have five days in a sense to make a decision.''
Overbay, 36, was cut earlier Tuesday by the Boston Red Sox, who were considering using the veteran as a reserve first baseman.
During the spring with the Red Sox, Overbay was 9 for 41 (.220) with a double, a triple and seven RBIs in 19 games. He flied out into a double play in his debut with the Yankees on Tuesday night.
"I believe there's an out Friday, so it's kind of like a three-day tryout," Overbay said. "There's not a lot of opportunities out there, so I wanted to jump on this. This was an opportunity, and that's all it was. Go out there and see what I can do, and who knows? Stranger things have happened."
Amongst the group auditioning to replace Teixeira, Overbay is considered a better defender than Rivera or Johnson. Rivera and Overbay possibly could platoon in Teixeira's absence.