Several reports say that player is Rangers second baseman Leury Garcia.
The White Sox had 72 hours to work out a trade for Rios after the Rangers claimed him off of waivers Thursday.
Rios, 32, is batting .277 with 12 home runs and 55 RBI this season. The Rangers had been looking for an extra right-handed bat since Nelson Cruz was suspended 50 games for his role in the Biogenesis scandal.
The Rangers have won four games in a row and are tied with the Oakland Athletics atop the American League West. CORCORAN: With Nelson Cruz suspended, Rangers hope pitching can save the day