The 31-year-old suffered the injury while washing dishes at his home last Saturday, and he's expected to resume throwing in approximately two weeks. A further determination will then be made regarding how long he'll be out following that stretch.
"This injury certainly challenges our depth in an area where we had some concerns already about our depth," Cubs president of baseball operations Theo Epstein said, according to Jesse Rogers of ESPN. "It's going to be important for certain players to step up and make the most of this opportunity."
The former All-Star posted a 4.68 ERA across 171 innings with the Cubs in 2019.
Quintana is scheduled to make $10.5 million in 2020. He's eligible to become a free agent following the campaign.