The two clubs had been discussing a deal for the playmaker worth 60m euros (£52.75m) - 55m (£48.35m) plus a maximum of 5m (£4.4m) add-ons.
Fekir had undergone a medical with Liverpool staff in Paris and agreed terms on a five-year-deal.
However, Lyon said on Saturday that talks about the potential transfer of their captain "had not succeeded".
In a statement released on the French club's official website, Lyon claimed that a breakdown in negotiations have resulted in French worldwide remaining where he is for the time being.
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Yet this high-profile club president has a history of dismissing transfer rumours surrounding his players shortly before they make a move, with the deal that saw Arsenal sign striker Alexandre Lacazette last summer and prime example of Aulas spinning media narrative to suit his own club's agenda.
A product of Lyon's successful youth academy, the skillful forward has 58 goals in 161 games for Lyon.
France forward Fekir has always been linked with a big-money move to Anfield and enjoyed a stellar campaign in France's top flight this season - scoring 18 goals and supplying a further eight assists.
Liverpool was hoping to add Fekir to their off-season signings of Monaco midfielder Fabinho and RB Leipzig's Naby Keita.
Lyon issued a statement on Friday to dismiss reports the 24-year-old was on the verge of signing for Jurgen Klopp's Champions League finalists. "There is something logical, it has already been done for others".