"We haven't made a final decision yet," Rangers general manager Jon Daniels said.
Both Byrd and Rodriguez are Type B free agents. That means if they are offered arbitration and sign with another club, the Rangers would receive a supplemental first-round pick in next year's First-Year Player Draft.
But the player could also accept arbitration, which would mean he becomes a signed player with his old team on a one-year deal.
The Rangers would welcome Byrd back. He hit .283 with 20 home runs and a club-leading 89 RBIs in 2009 and played well defensively in center field. The Rangers have expressed an interest in re-signing him, although little progress has been made to this point.
Rodriguez is a tougher decision for the Rangers. The Rangers acquired him on Aug. 17 from the Houston Astros after Jarrod Saltalamacchia went down with a shoulder injury, and Rodriguez was their starting catcher ahead of Taylor Teagarden for much of the remainder of the season.
Rodriguez, who turned 38 this offseason, hit .245 with two home runs and 13 RBIs in 28 games and has expressed a strong desire to finish his Hall of Fame career with the Rangers.
Texas remains committed to Saltalamacchia and Teagarden as its catchers. But the Rangers still want to see how Saltalamacchia recovers from shoulder surgery, and he is just now beginning to play winter ball in the Dominican Republic.
If Saltalamacchia is at full strength, having Rodriguez would only complicate matters. It would likely mean that Saltalamacchia is the starter, Rodriguez is the backup and Teagarden returns to Triple-A to play regularly.
If Saltalamacchia has trouble coming back from shoulder surgery, the Rangers would definitely need Rodriguez back to complement or compete with Teagarden.
Tuesday's decision could offer a clear indication how the Rangers feel about their catching situation.
The Rangers do not have a Type A free agent, but there are at least a couple of those that could interest them, including outfielder Jermaine Dye and relievers Darren Oliver and Octavio Dotel. If the Rangers sign a Type A free agent that has been offered arbitration, then they have to forfeit their first pick in the Draft as compensation to their former club.
Dye is not expected to be offered arbitration, as the White Sox already elected not to exercise a $12 million option. The Rangers have interest in Oliver, a one-time starter for them who has extended his career and flourished as a reliever for the Angels the past three seasons. The Rangers are unlikely to sign a Type A free agent if it will cost them a Draft pick.