Sanchez suffered the injury while working in the weight room Thursday as part of the team's strength and conditioning work.
The 29-year-old Sanchez, who is in his eighth NFL season, underwent minor surgery Saturday to repair a torn ligament in the left thumb. The timetable for his return is about a week.
The Broncos have a potential starter in Sanchez, who was acquired from the Philadelphia Eagles for a conditional seventh-round pick in March.
When OTAs begin May 24, Sanchez likely will miss the start of the workouts.
With Sanchez sidelined, second-year veteran Trevor Siemian and rookie Paxton Lynch, who was picked with the 26th selection in this year's draft, are expected to receive all of the work at quarterback during the first week of OTAs.
One day after he suffered his injury, Sanchez attended a prom with more than 100 patients at Children's Hospital Colorado.
Sanchez was a surprise guest at the hospital's 10th annual prom for patients on Friday, which was held at Sports Authority Field for the first time. Children's Hospital puts on the "Night of Champions" event each year for teenagers who are unable to attend their own proms because they are too sick.
Sanchez spent the evening visiting with the teens, signing autographs and taking photos.