The France international was on the verge of sealing an Anfield switch prior to last summer's World Cup.
But the Reds pulled the plug on the deal, citing fears over surgery Fekir had undergone on a knee injury earlier in the season.
Yet now Liverpool are back in for the midfielder, according to Gazzetta dello Sport's Nicolas Schira.
"Aulas wants at least €50m for his player.
"Liverpool have offered Fekir a contract until 2024 at €4m at year."
Lyon-based newspaper Le Progres claim the upcoming friendly between Lyon and Liverpool in the Swiss capital of Geneva on July 31 could play a part in negotiations.
It has been suggested that the clash could act as a sweetener for Lyon if they are pocketing the bulk of the match-day revenue.