Rios, who first injured the ankle July 19 in Toronto, missed the previous two days to get it some rest. But as he slid into the catcher's glove in the fourth, he came up hobbling again.
Rangers manager Ron Washington said Rios is day-to-day with the injury.
"It's about the same," said Rios, who had a RBI double in two plate appearances on Sunday. "It's something I have to deal with the rest of the season."
Rios was replaced in right field by Jim Adduci, who struck out twice.
Washington has two corner outfielders, Shin-Soo Choo being the other, with ankle problems. Choo has been dealing with a sprained ankle since April, and with Rios' health a daily question mark, it limits what the manager can do position-wise.
Washington wants Choo to either DH or play right field so Daniel Robertson and Adduci can get some playing time in left field. With Rios' status uncertain, it will be interesting to see what Washington does moving forward.