Andersen, 25, spent the 2020-21 season with the Cottagers and impressed under Scott Parker despite the west London outfit eventually being relegated back down to the Championship.
The Telegraph says his performances had attracted interest from Spurs, but Palace are now said to be the frontrunners.
The Eagles have already acquired Chelsea man Marc Guehi in the summer transfer window and Andersen would be another stellar addition to their squad as they look to move on from the tenure of Roy Hodgson.
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