With one out in the first, Franco lined a single. Nelson Cruz followed with a double, but Franco limped into third. After being examined by team athletic trainers, Franco left the game and was replaced at shortstop by Joey Wendle.
"We'll know more, but hopefully we caught it early enough,'' Rays manager Kevin Cash said. "I'm sure he'll miss some time, but we're hoping it isn't as bad as it could be.''
The 20-year-old Franco came into the game hitting .282 with seven homers in 61 games for the AL East leaders.
"I felt it when I pushed off second for third base. That's a play where I normally score,'' he said through a translator. "I had trouble with my hamstring in spring training (in 2019) and it was OK, so I hope I don't go on the (injured list)."
Franco's on-base streak is the second longest in major league history by a player under 21. Frank Robinson set the record with a 43-game string in 1956.
Information from The Associated Press was used in this report.