Texas claimed Rios after Chicago placed the outfielder on waivers. The two teams discussed a trade before the July 31 non-waiver trade deadline but could not reach an agreement. The White Sox asked for pitchers Luke Jackson and Martin Perez as well as infielder Rougned Odor, but the Rangers did not feel comfortable parting with all three.
The two teams have until Friday to complete a deal. The White Sox will then decide whether to pull Rios back or accept the Rangers final offer.
Rios entered Friday hitting .277/.328/.421 with 12 home runs. Texas is looking to replace right fielder Nelson Cruz, making Rios an obvious fit.
The Rangers offense, however, has picked it up after struggling in July, averaging six runs per game so far in August. Texas is also hopeful that Lance Berkman can return to the lineup in the next 10 days.