Carmona was 4-1 with a 2.25 earned-run average through his first nine starts before injuring himself and giving up six runs in two innings Friday night in a 13-9 loss to the Texas Rangers.
The Indians are expected to activate right-hander Jake Westbrook to replace Carmona.
The Colorado Rockies put left fielder Matt Holliday (strained hamstring) on the 15-day disabled list. He was hurt Saturday while running out a groundout in the ninth inning against New York. Outfielder Cory Sullivan was activated.
"It's very different than mine," Lester told the network. "His is slow growing. His is something that he's going to die with, not die from."
Lester, who threw a no-hitter last Monday against Kansas City, was diagnosed with a rare form of non-Hodgkin's lymphoma in 2006.