Fekir has long been tipped with a move away from the Ligue 1 club, and was hours away from completing a move to Anfield last year.
However, the forward's £53 million ($67m) deal fell through at the last moment , meaning he stayed put at Lyon for the 2018-19 season.
Fekir went on to net just nine league goals in 29 games in 2018-19, half his tally from the previous campaign, while Liverpool stormed their way to a 97-point Premier League season and won a sixth European title.
Former Lyon boss Genesio walked away from the club at the end of last term after pressure had mounted on his job, and he believes his former player should also move on.
"I think that he [Fekir] was very frustrated with his canceled transfer to Liverpool at the beginning of the season," Genesio told French radio station RMC.
"He made the World Cup, he had some injuries. He certainly did not feel strong enough to take on the role of leader, which he had done the previous season, relaying technical information on the pitch.
"Nabil is coming to the end of a cycle in Lyon. He has experienced a lot of things here.
"I think that he needs a new challenge, to push his limits, and to be self-critical. Because we know, we always need a personal challenge in a career, and to ask questions of ourselves."
But a move to Liverpool may no longer be on the cards.
Reds legend Neil Mellor has claimed Jurgen Klopp should not be signing the France international, claiming Liverpool need to reinforce other areas of their squad.
The Anfield club might be aiming to bolster their defensive options this summer having announced the signing of 17-year-old centre-back Sepp van den Berg from PEC Zwolle on Thursday.
And reports claim Dejan Lovren is keen to leave Merseyside after playing just 13 Premier League games last term.
The Reds have, however, been linked with a move to bring Philippe Coutinho back from Barcelona following his underwhelming performances at Camp Nou.