The 29-year-old striker scored two goals in 15 appearances last season as the Swans were relegated to the Championship.
Though Bony has expressed a desire to stay with the club, Swansea are determined to sell and are willing to accept an offer between £1.5 million and £2m.
The Welsh side have been shopping Bony to a number of clubs, but Galatasaray appear to have moved to the front of the pack as their interest has grown in recent days.
Bony was prolific in his first spell with the Swans, scoring 26 times in a season and a half between 2013 and 2015.
Sold for £25m to Manchester City in 2015, the Ivorian struggled to reproduce the form he had shown with the Swans, scoring six league goals before moving on loan to Stoke on 2016.
The Ivorian returned to Swansea ahead of the 2017-18 campaign, but could not prevent the club from being relegated after seven years in the top flight.