Connaughton suffered the injury during the third quarter while committing a shooting foul on Suns guard Chris Paul. He left the game with 3:34 remaining in the third and did not return.
"We're trying to figure out who will see him next, what the next step is," Bucks coach Mike Budenholzer said on a video call after the game. "Unfortunate, but we're hopeful that he'll be able to return. When that is, not exactly sure. But we'll see."
Connaughton was averaging 10.2 points and shooting 41% from 3 (both career-highs) in 53 games for the Bucks this season. He has moved into a bigger role this year as Milwaukee's sixth man and occasional spot starter, playing a career-high 27.2 minutes per game.
"He plays a huge and key role for us coming in the starting lineup every now and then," said Bucks forward Khris Middleton, who said he saw Connaughton wearing a cast after the game. "Just being that glue guy for us on both sides of the ball. It's going to be tough without him for a while so hopefully he can get back sooner than later."
Earlier Thursday, the Bucks made a move to acquire forward Serge Ibaka from the Clippers to bolster its frontcourt, but in the process, Milwaukee traded away its depth on the wings.
Donte DiVincenzo was traded to the Kings as part of a four-team deal earlier Thursday afternoon. Rodney Hood and Semi Ojeleye were sent to the LA Clippers. Their remaining options to replace Connaughton include George Hill, who has been out since Jan. 28 with a neck injury, and Wesley Matthews, who has struggled since being signed by the Bucks midseason.
"That's how it goes sometimes," Bucks guard Jrue Holiday said after Thursday's game. "I don't know if it's ironic or whatever you want to call it. Definitely unfortunate. Sometimes that's how it works."