Jurgen Klopp is looking to make the former Bayern Munich winger their his new signing of the summer.
Shaqiri has a release clause which means he can leave Stoke for around £13.5million after their relegation.
The Reds have opened discussions with the Potters, who are resigned to losing the Switzerland international.
Shaqiri has been considering his options since his involvement in Russia was ended at the last 16 stage by Sweden last week.
The Liverpool ECHO understands that the 26-year-old, who is currently on holiday, favours a move to Anfield and Stoke expect him to complete the switch to Anfield.