Lowrie was still feeling lightheaded Sunday before the series finale at Texas. He didn't play Saturday night after coming out of Friday night's game due to heat exhaustion.
Lowrie was removed from Friday's game in the eighth inning after feeling dizzy and nauseous. He needed two IV bags to rehydrate and said he felt better on Saturday.
Manager Terry Francona told Lowrie to check in with him Sunday morning just to make sure he was OK to play.
"He woke up this morning and didn't feel good," Francona said. "He got to the ballpark and rallied a little bit. We sent him down to the cage. He swung the bat OK, but he was lightheaded. He's going to get a bunch of fluids and see if he can be available for later."
Lowrie said he felt like he was playing in a sauna during Friday's game.
Lowrie missed the first 94 games this season on the disabled list with mononucleosis. Francona says it is "common sense" to think the illness might have left Lowrie vulnerable to the heat.
The temperature at the start of Saturday night's game was 102 degrees, the second-hottest ever at Rangers Ballpark. For the day game Sunday, it was already 100 degrees more than an hour before the game, and expected to reach about 104.
Information from ESPNBoston.com's Joe McDonald and The Associated Press contributed to this report.