The Premier League club are understood to have offered around €34 million (£30 million) for the 18-year-old, but the proposal was turned down immediately.
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According to transfer expert Nicolò Schira, Atalanta value Ahanor at between €40 million and €45 million, meaning Brighton would need to significantly raise their offer to progress negotiations
Ahanor only joined Atalanta from Genoa last summer for €20 million, making him one of the most expensive Italian teenagers in history.
He enjoyed a strong breakthrough season, featuring 22 times in Serie A, as well as nine Champions League appearances and four in the Coppa Italia.