The report claims that the Reds are targeting bringing in a new goalkeeper over the winter transfer window.
Manchester United had been closely linked with Muslera over the summer as a potential replacement should David de Gea had left for Real Madrid.
Muslera played a key role in Galatasaray winning the league for a record 20th time last season and earning the right to wear a fourth star – Galatasaray also won the double for a record sixth time.
The 29-year-old also played a key role in Galatasaray’s Turkish Super Cup win over Bursaspor. Muslera has won three league titles at the Lions and also the Copa America with Uruguay.
The experienced keeper has been one of the Istanbul giants most influential players over the past few seasons and still has another two-years left on his contract but Galatasaray could be tempted to part ways with Muslera to reduce the bill.