The Blues have agreed a deal with the Italian side worth a reported £40 million, per BBC Sport.
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He will officially become a Chelsea player next summer following his loan spell with Palace.
On the move, Ahanor told Palace's club website: "I'm very happy to be here and to be a part of this great family.
"To be at one of the best clubs for young players in the Premier League is something which makes me grateful and excited."
The highly-rated Italian youngster made 28 appearances for Atalanta last season, and became a regular starter for the club.