Rodriguez, 38, will serve as a backup and mentor to Jesus Flores, who is expected to be ready for Spring Training after having surgery on his right shoulder and right elbow, according to general manager Mike Rizzo.
Rizzo didn't mention Rodriguez during his Monday session with the local media, but he spoke about the type of backstop he wanted.
"We want a catcher that could fill in as an everyday guy on the short term, but could morph into a mentor/teacher for Flores," Rizzo said. "We have been consistent with that all the time. We are still looking for that guy. We are not sure if he is out there, but we are really checking into it."
Rodriguez has played in 14 All-Star Games and won 13 Gold Gloves, most by a catcher and tied for fourth most at any position. The 1999 American League MVP Award winner also has more runs (1,277), hits (2,618) and doubles (526) than any other backstop in Major League history.
Rodriguez served as a backup for the Rangers after being acquired from the Astros in August. He played in a combined 121 games and hit .249 with 10 home runs and 47 RBIs.
The move means that catcher Wil Nieves, who is arbitration-eligible for the first time this season, will be non-tendered. With Flores on the disabled list for most of the 2009 season, Nieves platooned with Josh Bard and hit .259 with one home run and 26 RBIs.
The Nationals had also expressed interest in Brad Ausmus earlier Monday to serve as a backup/mentor for Flores.