The 26-year-old left the field hurt with five minutes remaining in the second quarter of the 34-20 loss to Baltimore Ravens on Thursday.
Burrow bent over in pain after throwing a touchdown pass to Joe Mixon.
Bengals head coach Zac Taylor said he had a torn ligament in his right wrist.
Burrow spent several minutes in the medical tent before jogging off the field to the locker room for further treatment and television images showed swelling around his wrist.
The NFL is investigating the team's reporting of Burrow's injury status before the game.
Burrow said he wore a compression sleeve on his wrist but was not injured until Thursday.
"This is a completely different thing," he added. "It's not uncommon for guys to wear compression sleeves on the plane.
"When you go up to that altitude, things can swell up. This is a completely new injury."
Jake Browning took over at quarterback for the Bengals, who are now 5-5 for the season and bottom of the competitive AFC North division.
The Ravens, who are now 8-3, lead the AFC North, with quarterback Lamar Jackson throwing for 264 yards and three touchdowns in his team's victory.
Since Burrow was drafted with the first overall pick in 2020, the Bengals have a 2-5 record when he has not started.
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