Liverpool remain confident, despite Lyon president Jean-Michel Aulas claiming the deal is at a "standstill".
However, L'Equipe have offered Liverpool fans a boost before the global tournament that kicks off next week, with the French outlet claiming that a deal is very close to being complete.
Fekir will be given permission to leave France's World Cup training base and could undergo his medical on Merseyside on Friday.
According to French newspaper L'Equipe, the Reds have agreed a deal worth £57 million (€65 million) for the 24-year-old and he will have a medical on Friday.
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Klopp and co. hope to complete the transfer before the commencement of the FIFA World Cup, next week.
Fekir, who can play in both midfield or attack, is set to fill the creative void left by the January departure of Philippe Coutinho to Barcelona. Fekir is set to play for France on Saturday evening against the US, a game that will take place in Lyon's Stade des Lumieres.
A fee has yet to be revealed, although it is rumoured that Liverpool will have to pay around €65m (£57m approx) to make Fékir their second summer signing.
Fekir will be the third new player to arrive at Anfield this summer, behind Monaco man Fabinho who was signed earlier this window, and RB Leipzig's Naby Keita, whose signature was acquired last summer.