Hayward will visit a hand specialist Monday to determine whether surgery is needed, his agent, Mark Bartelstein, told ESPN's Adrian Wojnarowski.
The 29-year-old sustained the injury while colliding with a screen set by Spurs big man LaMarcus Aldridge toward the end of the second quarter. Hayward left the game and did not return.
Two years removed from a devastating ankle injury that wiped out all but five minutes of his 2017-18 campaign, Hayward has enjoyed a return to his previous All-Star standard. Entering Saturday's contest, the Butler product had averaged 20.3 points on 56.4% shooting - including 44.4% on 3-point looks - to go along with 7.9 rebounds and 4.6 assists per game this season.
The Celtics improved to 7-1 on Saturday, which is tops in the Eastern Conference.