"It looks like a serious knee injury, but how serious it turns out to be after he has the scan, we will have to wait and see," Hodgson said about Dann's knock.
"You don't know directly how bad it is after the game and the doctors loathe to make any bold statements but it doesn't look good."
Palace became only the second team to take points off City in the league this season, halting their winning streak at 18 games, and travel to 13th-placed Southampton on Tuesday but face a tough task with a stretched squad.
"Mamadou Sakho has been out now for a month and is likely to be out for a few more weeks to come with a quite serious injury, and Joel Ward is still recovering," Hodgson said.
"After an effort like this you'd wish we could make the changes Manchester City feel they can always make when you look at the quality of the players they have who aren't playing."
Palace are 17th in the league with 19 points after four wins, seven draws and 10 defeats.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by Sudipto Ganguly)
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