The Rangers received cash, and no player, in exchange for Soto. Texas scratched Soto from the lineup before Sunday's game with the Kansas City Royals.
Soto started the season on the disabled list as he underwent surgeries to repair an ankle and a knee. When he finally returned, he suffered a right groin strain running the bases in New York and was placed on the 15-day disabled list.
In 10 games, Soto hit .237 with one home run and three RBIs.
J.P. Arencibia is the emergency catcher for Sunday's game.
The team most likely will add another catcher, and could call up Triple-A Round Rock's Tomas Telis, 23, who is hitting .318 in 106 minor league games this year.