According to Sky Sports, the Ligue Un side are currently offering just £5million for the midfielder, who is valued at more than double that fee.
Negotiations between the two clubs are ongoing as they hope to reach an agreement that suits all parties.
Lyon are keen to strengthen their attacking options after losing star man Memphis Depay on a free transfer to Barcelona.
They also sold Danish centre-back Joachim Andersen to Crystal Palace for an initial fee of £13.7million last month.
Shaqiri has played infrequently for Liverpool over the last three years, often being used as an impact substitute since arriving from relegated Stoke City.
With an established forward line of Mohamed Salah, Sadio Mane and Roberto Firmino, he has rarely featured in Jurgen Klopp’s starting line-up.
But Shaqiri has still made some notable contributions to the Liverpool cause, particularly during his first season, when he chipped in with six goals.
In total, he has made 63 appearances for the club, lifting the Champions League, the Premier League, the Super Cup and the Club World Cup during his time at Anfield.
Shaqiri’s current deal at Liverpool expires next summer, but the club have the option to extend it by a further year.
Once more faced with limited opportunities to play, he might choose to move elsewhere in search of regular first team football.
There is little doubting Shaqiri’s technical ability, which he showcased impressively throughout Switzerland’s run to the quarter-finals of Euro 2020.
He scored a brace in the 3-1 win over Turkey that secured their progress the knockout rounds and then grabbed an equaliser against Spain, who eventually prevailed on penalties.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.