The London Evening Standard says Fulham's bid exceeds the £34m spent to land Emile Smith Rowe from Arsenal last year.
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However, the offer is said to fall short of Shakhtar's €50m asking price.
Fulham are now prioritising a deal for Kevin ahead of bringing Reiss Nelson back from Arsenal after last season's loan.
Kevin, 22, was impressive for Shakhtar last season, catching the eye of scouts across Europe during their Champions League campaign.
