French football website Le Parisien, as quoted by Get Football France, is reporting that the Reds have submitted an offer thought to be worth just below €60m (£52.65m) for Lyon captain.
The same article states that Fekir and his representatives are still hesitant about a potential switch to the 18-time English champions.
According to the same story, Lyon are in no rush to sell the attacking midfielder before the World Cup despite Liverpool FC being linked with Fekir over the past few weeks.
Indeed, Le Parisien suggest that the French club would rather wait until after Russia 2018 in the hope that Fekir’s performances will spark a bidding war and push his price up.
Liverpool FC have already been active in the summer transfer market after the Reds completed a £40m deal to sign Brazil international Fabinho from AS Monaco earlier this week.
The Reds finished in second place in the Premier League table to secure a spot in the Champions League in successive seasons under Jurgen Klopp.
However, Liverpool FC lost 3-1 to Spanish giants Real Madrid in the Champions League final in Kiev last weekend.
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