Smart, the 2021-22 NBA Defensive Player of the Year, was dealing with a right quad contusion and then took two shots to the quad on back-to-back plays in the third quarter.
"He's a tough guy who's going to try to play through things," Udoka said. "We'll get him some treatment today and see how he feels tomorrow."
Boston's official injury report will come out later on Monday.
When Smart exited the game in the third quarter, he needed to hit the exercise bike before checking back in with 8:37 left in the fourth quarter. He played until Udoka took out his starters with 1:59 to go in the 101-89 loss.
Udoka said it's more than just a tolerance issue with Smart, who also missed six games in January because of a quad issue.
"It's something that's going to be stiff. He got hit twice there," Udoka said. "He's had this in the past obviously where he missed games this year. So it's more than just pain tolerance. It's tightness and restriction of movement as well.
"We'll spend today getting treatment and in the morning get some more treatment and see how he feels at game time."
Smart also had to leave the game with 1:12 remaining in the second quarter on Sunday because of a stinger that left him walking off the court while dangling his right arm. Udoka said that wasn't an issue for him on Tuesday.
In 33 minutes, Smart finished with 10 points and six assists.
Jaylen Brown entered Sunday's game with a lingering hamstring issue but Udoka said he didn't think that limited Brown despite his 4-of-13 shooting performance.
"I think it was more so a product of how they were guarding us," Udoka said. "We had some good looks from 3, some wide-open looks he got there. Then attacking the basket you didn't see many limitations as far as that. He felt good coming out of the game."