The Cottagers have been attempting to strike terms with Shakhtar for the Brazilian over the past two months.
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BBC Sport says less than 48 hours away from the closure of the summer market, a deal is now close to being reached.
Fulham are willing to pay a club record €40m to convince Shakhtar to sell.
Kevin has scored 17 goals and provided 10 assists in 57 matches for the Ukrainian side.
For the moment, former Montpellier goalkeeper Benjamin Lecomte is Fulham's lone summer signing.