Rios, who is hitting just .231 over his last seven games, originally injured his ankle coming out of the batter's box July 19 at Toronto and missed the Rangers' next three games.
"It hurts and it hasn't been getting better at all," Rios said. "It's kinda tough I could play on it, but I'd rather take a couple of days off to get it right."
Rios said he's available to pinch hit and play defense if manager Ron Washington needs him to.
He could return to the lineup on Sunday.
"I'm going to be in the lineup, no matter what," Rios said. "I just want to give it a rest and see if it can feel better."
Rios cleared waivers Thursday and can be dealt to any club, but his injury could put potential talks on hold for now. Rangers officials have said they are open to picking up Rios' team option for 2015, but GM Jon Daniels said the team will look at trades if some quality minor-league prospects can be acquired.
Soto, who has played in only three games this season, hit .375 in a five-game rehab stint at Triple-A Round Rock. Williams was 1-1 with a 9.90 ERA in his two starts with Texas.
Notes: Outfielder Jake Smolinski was placed in a walking boot Wednesday because of a small fracture in his left foot and will be out two-to-three weeks. Smolinski bruised left foot July 21 and went on the DL on July 25, retroactive to July 22.