The winger, who scored seven goals for the newly-relegated Potters last season, has a release clause of £13.5m.
Following Stoke’s relegation to the Championship, the 26-year-old expressed a desire to remain in the Premier League, recently stating it is “no secret” he will leave the bet365 Stadium.
A move to Anfield would make Shaqiri the Reds’ third signing of the summer following the arrival of midfielders Fabinho and Isaac Christie-Davies.
Shaqiri joined Stoke in 2015 from Inter Milan before going on to make 84 league appearances for the club.
One of Switzerland’s star performances in Russia, his last-minute winner against Serbia sent the side through to the World Cup’s round of 16 before their campaign drew to a close with a defeat to Sweden.
The Swiss’ contract at Stoke runs until 2020.