The 38-year-old Payton, in his 16th NBA season out of Oregon State, suffered the injury Monday in practice.
The nine-time all-star is averaging 5.3 points and nearly three assists per contest in 67 games (27 starts) primarily as a backup point guard to Jason Williams.
His career averages are 16.9 points, 6.9 assists, and 4.1 rebounds in 1,267 NBA games.
Jason Kapono, who has missed 14 games with a sprained ankle, likely will see more time at guard in Payton's absence.
Payton's playing time would have declined because of the return of Dwyane Wade, who missed nearly two months because of a dislocated left shoulder.
Payton also has been slowed by a right groin injury in recent weeks.
His absence begins with Wednesday night's game against Washington.