According to reports, Sow is now their top target and the Merseysiders have made an initial £10m offer for his services.
It is unclear as to whether this will tempt the Istanbul club to sell, but with the Senegalese striker having been relatively disappointing since his 2012 arrival, they may be willing to cash in.
The 26-year-old reportedly turned down the opportunity to join Arsenal before switching to Fener.
He was also linked with Liverpool, and cited that it would be a “dream” to play for the club, which would suggest that a deal could be concluded.
Sow had an impressive spell in France with Lille, before moving to Turkey, where he was a regular scorer, averaging a goal every other game.
With only Luis Suarez and Sturridge as Brendan Rodgers’ available strikers, Liverpool are said to be keen to add further bodies to their squad.
They have also been linked with Paris Saint Germain’s Kevin Gameiro and Barcelona star David Villa, both of whom could be available on loan deals.