Inter president Beppe Marotta has confirmed their interest in the Frenchman on multiple occasions since the end of last season.
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Transfer expert Fabrizio Romano is reporting a firm offer of €24m has now been lodged.
Inter's previous offer of €22m plus bonuses was rejected by Parma earlier this week.
Nerazzurri management see Bonny, 21, as ideal cover for Marcus Thuram, particularly with Marko Arnautovic leaving this summer.