Texas also got an international signing bonus pool slot worth $250,000 in the deal while sending minor league infielder and outfielder Jonah McReynolds to the White Sox.
Castillo's deal includes a $500,000 buyout for 2020. The 32-year-old catcher hit .209 over 72 games last season -- his second with the White Sox and 10th in the majors.
The 23-year-old McReynolds hit .239 in 56 games with Advanced Rookie Spokane.
Chicago reinstated pitchers Michael Kopech, Ryan Burr and Carlos Rodon and outfielder Jon Jay from the 60-day injured list on Thursday. Slugger Jose Abreu, Jay and pitchers Ross Detwiler, Ivan Nova and Hector Santiago declared for free agency. And lefty Josh Osich was claimed off waivers by Boston after being outrighted to Triple-A Charlotte.