The 22-year-old arrived at the Turkish giants from Rennes for just over £1m last summer, but an injury-hit first season at the club saw him feature only 19 times across all competitions.
Boey has since started in all eight of Galatasaray's Super Lig fixtures this term, establishing himself as a mainstay in the starting lineup.
A recent report from Turkish news outlet Fanatik claims that Liverpool sent scouts to watch Boey play against Konyaspor last month.
According to Takvim, Galatasaray will only entertain offers in the region of €15m (£13m) for the right-back, who is under contract with the Turkish club until the summer of 2025.
The report adds that Liverpool could face competition from French trio Rennes, Marseille and Lyon for the player's signature when the transfer window re-opens in the New Year.
Boey is believed to be one of four Galatasaray players who Liverpool are closely monitoring, with a recent report claiming that Victor Nelsson, Kerem Akturkoglu and Yunus Akgun have all been identified as potential targets.
Twenty-three-year-old centre-back Nelsson is a Danish international who, like Boey, joined Galatasaray last summer and is now regarded as a first-team regular in defence.
Akturkoglu, also 23, enjoyed a prolific spell in front of goal for Galatasaray last season, recording 13 goals and 13 assists in 52 games across all competitions, and the Turkish winger has since found the net twice in eight appearances this term.
Both Nelsson and Akturkoglu are under contract until June 2026, while 22-year-old winger Akgun - who is currently on loan at Adana Demirspor - will see his deal expire in 2024.
Jurgen Klopp bolstered his squad with the additions of Darwin Nunez, Fabio Carvalho, Calvin Ramsey and Arthur Melo over the summer, with the former arriving at Anfield for a fee which could rise to a club-record £85m.
Following a poor start to the 2022-23 campaign, Liverpool are believed to be keeping a significant sum of cash aside ahead of spending big in the New Year.
As well as Galatasaray's quartet, the Reds have also been linked with the likes of Real Betis' Guido Rodriguez, Rennes' Jeremy Doku, and Paris Saint-Germain's Kylian Mbappe.