Naylor sat out the last two games in Houston with the injury that had been bothering him for longer. An MRI revealed the strain, which has a projected recovery time of 3-6 weeks.
"The problem with these is it's not like a calf or a quad, where if you're starting to feel good, you could play through it," manager Terry Francona said. "We're going to have to keep an eye on him and he'll get re-checked weekly."
The Guardians, who enter the weekend 2½ games behind first-place Minnesota in the American League Central, made the move with Naylor -- retroactive to Tuesday -- on Friday before opening a three-game series against the Chicago White Sox.
Naylor has been Cleveland's most productive middle-of-the-lineup hitter. He's batting .306 with 15 homers and a team-high 79 RBI.
Earlier this week, the Guardians traded slugger Josh Bell to Miami.
The Guardians also placed infielder Tyler Freeman on the injured list with a strained right shoulder. Francona said Freeman's injury is not a long-term concern.
Needing some immediate help, Cleveland acquired outfielder Kole Calhoun in a trade from the Los Angeles Dodgers for cash and recalled infielder José Tena from Triple-A Columbus.
Calhoun, 35, split the season in Triple-A with the New York Yankees and Dodgers. He batted a combined .297 with nine homers and 46 RBI in 58 games.