Kessie, 20, is said to have attracted interest from across the continent after impressing in Serie A.
Manchester United are reportedly tracking the Ivorian, who joined Atalanta in 2015, while Chelsea have been strongly linked.
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According to the report, Everton made a £15.6million (€18m) bid plus bonuses for Kessie – but their offer was rejected.
Paris Saint-Germain’s £21.6m approach has also been turned down, while Atalanta have snubbed interest from Chelsea boss Antonio Conte in the midfielder.
Reports earlier this week claimed Roma had launched an effort to sign Kessie, with Manchester United willing to spend £26m on the youngster.
Everton reportedly had a £15.5m bid for Frank Kessie rejected
Kessie is currently on Africa Cup of Nations duty with Ivory Coast
Chelsea, Manchester United, Roma and PSG have also been linked with Kessie
But Atalanta are determined to keep Kessie after already letting Roberto Gagliardini go on loan to Inter Milan.
But the report claims they may struggle to prevent Kessie’s departure should the bidding continue to rise.
Gazzetta claim Juventus have contacted Kessie’s agent regarding a possible deal, while Napoli and Roma are also keen on the burgeoning star.
However, it appears as though Atalanta will retain Kessie until the end of the season, when they will listen to a growing number of offers from Italy, France and England.