The Rangers have been watching Pierzynski for the past three weeks, according to the source.
The catcher was a major catalyst for an up-and-coming White Sox team that won the World Series in 2005. The Rangers envision Pierzynski being that type of catcher for them with a young pitching staff.
Pierzynski is in the last year of a $6.7 million contract. He will become a 5-and-10 man on June 14. Any major league player who has 10 years in the majors and five with the same team has veto rights on any trade.
Pierzynski has struggled, along with his teammates, at the plate this season. He's hitting .198 with two home runs and 11 RBIs. He hit .307 with 49 RBIs in 2009.
The Chicago Tribune was the first to report the Rangers' interest.
Bruce Levine covers baseball for ESPNChicago.com.