The Anfield club have long been linked with a move for the midfielder and have until Thursday afternoon to complete a deal.
Liverpool came within hours of announcing the transfer before this summer's World Cup but pulled out of the switch due to concerns about Fekir's medical.
He went on to win the World Cup with France but appeared set to remain in his homeland.
But according to daily newspaper L'Equipe, Lyon would allow Fekir to depart if they receive an appropriate offer - believed to be upwards of £60million.
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