ESPN's Adrian Wojnarowski reported on Sunday that Nance was expected to miss four to six weeks.
Nance played Saturday against the Milwaukee Bucks after missing the previous three games with a sprained right wrist. ESPN's Brian Windhorst noted that he fractured his finger during the Milwaukee game.
The 28-year-old has appeared in 19 games with 18 starts this season amid the Cavaliers' 10-14 start. He owns averages of 9.3 points, 6.8 rebounds, 3.2 assists and 1.9 steals per game, while shooting 48.3 percent from the field and 38.1 percent from beyond the arc.
The Wyoming product, who is in the middle of a four-year, $44.8 million deal with the Cavs, previously missed eight games in the 2018-19 season due to a right knee sprain and started much of last season on the bench.
While Nance doesn't put up monster numbers, he is a do-everything player who contributes in multiple areas and at both ends of the court.
There will be even more pressure on star center Andre Drummond to dominate until Nance is able to come back, while the rest of the frontcourt depth will be put to the test as well.
Former Brooklyn Nets Taurean Prince and Jarrett Allen could both see action at power forward alongside Drummond, and veteran center JaVale McGee could receive increased playing time as well.