Shaqiri has found it hard to clock up game time on a regular basis under Reds boss Jurgen Klopp and was linked with a move away from Anfield during the recent window.
The winger is yet to start a single Premier League game this season and Galatasaray are aware of his situation.
And according to Turkish daily Fotomac, Galatasaray coach Terim watched Shaqiri in Nations League action and has asked to club to check a swoop for him.
The Turkish club believe that the winger would not refuse a move if an agreement was reached with Liverpool and would be looking for a loan deal.
Shaqiri is keen to play as much as he can ahead of Euro 2020 and is likely to find game time tough to come by if he stays at Liverpool.
The former Stoke City man has regularly been linked with a move to Turkey in recent years, but has yet to sample the country’s Super Lig.
Shaqiri has so far registered 46 appearances in all competitions since he arrived at Anfield from Stoke in the summer of 2018.