The Rays said a timeline for Glasnow's possible return would be determined after he undergoes further medical examination.
Glasnow said he won't have surgery for now, and instead will try to strengthen the area. He plans to see another doctor on Friday.
If Glasnow chose to undergo Tommy John surgery, that would sideline this season and could prevent him from pitching next year, too.
Glasnow, 27, was placed on the 10-day injured list, and outfielder Mike Brosseau was recalled from Triple-A Durham.
Glasnow left Monday night's game against the Chicago White Sox after four innings with right elbow inflammation.
He allowed two runs on three hits and exited with Tampa Bay leading 3-2 in a matchup of teams that entered with the two best records in the majors. The 6-foot-8 right-hander walked one and struck out six, throwing 40 of 53 pitches for strikes.
Speaking after the Rays' 5-2 win, Glasnow said he sensed something wasn't right, but not terribly painful, in the elbow as he tossed his final pitches in the fourth. He told the coaching staff and was replaced by Ryan Thompson.
"The last couple of pitches I felt just a little tug,'' Glasnow said. "I didn't want to go out and chance it. The velo was still there; it just felt not right.''
Glasnow is 5-2 with a 2.66 ERA this season and is 10-2 over his past 21 starts dating to Aug. 12, 2020. He has 123 strikeouts in 88 innings while walking just 27 batters.
The Associated Press contributed to this report.