Darvish was scheduled to pitch on Thursday against Tampa Bay, but told the team on Monday that he felt discomfort in the elbow playing catch.
He said it didn't bother him on Saturday, when Darvish was roughed up by the Houston Astros, throwing 113 pitches in four innings and allowing six runs -- five earned -- and nine hits.
Darvish, who finished second in the American League Cy Young voting last season, is 10-7 with a 3.06 ERA and 182 strikeouts this year.
Darvish was examined on Tuesday by doctor Keith Meister, who didn't think there was ligament damage in the elbow. But the team said in an article posted on its website that Darvish would have a precautionary MRI exam.
"Yu expressed he could pitch if we were in a different spot," Rangers general manager Jon Daniels told MLB.com. "If we were in a more meaningful spot (in the standings) Yu would be good to go. But he agreed and we agreed to be cautious and smart about it."
However, Daniels said it was "premature" to talk of shutting Darvish down for the season.