The 24-year-old has been heavily linked with a move to Jurgen Klopp's side in recent weeks, with these talks suggesting a move is a distinct possibility.
Fekir is enjoying the best season of his career with Lyon, having scored 23 goals and laid on seven assists in 40 games in all competitions.
According to a report in 'L'Équipe' on Monday: "Nabil Fekir will leave Lyon this summer. The interest of Liverpool has grown and the player is not immune to it.
"The Reds should contact Olympique Lyonnais to enter discussion after the Champions League final, in which they face Real Madrid on Saturday."
It appears Fekir will be the only player that is allowed to leave Lyon this summer as they look to avoid falling into the same trap as Monaco, who suffered a significant exodus of talent last summer.
The club are aware of the fact that Fekir has outgrown them and do not wish to stand in his way, providing the price is right.
Asked recently if he had played his last game for the club Fekir replied: "Maybe. Frankly, I don't know. I spent an exceptional season here with the player, the fans and the staff. It ended beautifully.
"You have to pass a course, it's true. I think I gave everything in Lyon. Now it's time to look into the future."