The 30-year-old leadoff hitter sustained the injury while trying to steal second in the ninth inning of the Indians' game on Sept. 29 in Minnesota. Cleveland clinched an AL wild-card playoff spot with a 5-1 win.
Bourn played three days later in the wild-card game against Tampa Bay, but went 0 for 4 and struck out twice.
He batted .263 with six homers and 50 RBIs in his first season with the Indians, who signed him to a four-year, $48 million contract in February.