ESPN's Buster Olney noted Kluber exited after he was hit in the right elbow by a 102 mph line drive off the bat of Brian Anderson.
Paul Hoynes of Cleveland.com reported Kluber suffered a non-displaced fracture of his right ulna and will be re-evaluated on Thursday.
Consistency and durability have been hallmarks for the veteran right-hander since he had his Cy Young breakthrough in 2014. According to FanGraphs, he ranked second in innings pitched (1,091.1) and fifth in FIP (2.84) between 2014 and 2018.
While Kluber's injury is obviously a concern for the Indians, Cleveland has other talented options in its starting rotation. Trevor Bauer finished sixth in the American League Cy Young voting a season ago, and Carlos Carrasco is coming off an excellent year as well.
Assuming Kluber will be placed on the injured list, the Indians could turn to Cody Anderson as a spot starter. Injuries have hit a number of Cleveland's starting pitchers over the past few years, so shuffling the rotation isn't an unfamiliar spot for manager Terry Francona.