The 25-year-old was strongly linked with a move to Anfield ahead of the 2018 World Cup, only for the Reds to pull out of a deal at the final moment due to concerns over his long-term knee issue.
Lyon president Jean-Michel Aulas recently confirmed that Fekir would be leaving Les Gones this summer.
According to Sport, both Liverpool and City have submitted bids for the France international, who scored 12 times and contributed nine assists in 39 appearances for his Ligue 1 side during the 2018-19 campaign.
The report has not named a price for either offer, although it is thought that Fekir will be available for a cut-price fee this summer as his contract expires in June 2020.
Real Madrid have also previously been credited with an interest in the attacker. body check tags ::