The winger is understood to have a £13.5m release clause in his contract due to Stoke's relegation from the Premier League, with the winger ready to leave the club rather than spend the next season in the Championship.
Jürgen Klopp is thought to keen on signing Shaqiri as a quality back up to the front three of Mohamed Salah, Roberto Firmino and Sadio Mane.
Shaqiri, 26, recently expressed his desire to remain in the Premier League and missed just two of Stoke’s league matches in their relegation season.
“Of course, I would like to play on the biggest stage in football,” he said.