Dalton was injured in the third quarter of Cincinnati's 35-20 loss to the Cleveland Browns on Sunday. Jeff Driskel, who replaced Dalton on Sunday and completed 17 of 29 passes for 155 yards and one touchdown, is likely to start this week.
The Bengals now have 13 players on injured reserve.
Dalton completed 226 of 365 passes for 2,566 yards, 21 touchdowns and 11 interceptions in 2018.
Dalton last missed a game in 2015, when he broke the same thumb but avoided surgery.
He was injured Sunday attempting to recover an errant snap that sailed over his head and was recovered by the Browns. Dalton left the field and immediately went to the locker room.
Cincinnati also claimed Tom Savage off of waivers from the 49ers.
Savage is 2-7 as a starter, including a win in his first NFL start with the Houston Texans over the Bengals and Dalton on Dec. 24, 2016.
--Field Level Media
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