France, 27, exited Thursday's 2-1 victory over the Oakland Athletics because of the injury but said after the win that he was hoping to avoid a stint on the IL.
While running to first base, the Athletics' Sheldon Neuse hit France's left arm as he extended to catch a throw from shortstop J.P. Crawford in the bottom of the fifth inning. France immediately fell to the ground and clutched the arm in pain. He walked off the field on his own power.
Seattle manager Scott Servais said X-rays were negative but that a Friday MRI showed the move was needed.
"I heard that crack, and the first thing I thought was a torn UCL, dislocated elbow or shoulder," France said. "You think about being out for the rest of the year and needing surgery. A lot of negativity popped into my mind, but the MRI revealed just a strain. It's now about calming down the inflammation."
France was on track for All-Star Game consideration. He is seventh in the American League with a .316 batting average and is tied for the major league lead with 25 multihit games. He also has 10 home runs and 45 RBIs and has been hit by 14 pitches.
Servais said there are many options to play first base in France's absence, adding, "It will be matchup based and keep rotating it, through I'm not sure how it is all going to play out."
Seattle also announced before the game that it had acquired outfielder Trey Amburgey in a trade with the Cincinnati Reds. Amburgey had played in 39 games with Triple-A Louisville this season.
The Associated Press contributed to this report.