The Lions have made strengthening central midfield a priority and Inler is one of the names on their shortlist.
Galatasaray also wanted the Switzerland international last season before his move to Leicester.
The Istanbul giants would be able to offer Inler first-team football but Galatasaray will not play in the Champions League this season due to serving a one-year European competition ban due to breaching FFP regulations.
Despite being part of the Premier League winning Leicester side Inler struggled for playing time and is not expected to feature regularly next season either.
Inler would be able to play under domestic status in Turkey due to being of Turkish origin.
The experienced midfielder was born in Switzerland to Turkish parents and grew up in Turkey before returning to his country of birth.
Galatasaray are also in talks with Liverpool over singing Brazilian midfielder Lucas Leiva.