The injury reportedly occurred when a puck hit Connolly's right hand during practice.
Prior to joining Boston, Connolly spent four years with the Tampa Bay Lightning, who drafted him with the No. 6 pick in 2010.
Connolly, 22, has recorded 12 goals and three assists over 50 games this season.
The Bruins currently sit in the second Wild Card spot in the Eastern Conference with a 31-22-9 record.