The Turkish giants have formally declared that they are attempting to sign the prolific striker, while the completion of Nordin Amrabat's transfer is close at hand
The 26-year-old striker is one of Turkey's most sought-after players after an impressive season in which he topped the Super Lig's scoring charts with 33 goals.
A statement issued by Galatasaray read: "Formal negotiations with professional football club Trabzonspor for the transfer of Burak Yilmaz have begun. The public will be informed of any further developments."
Burak, who has 18 Turkey caps to his name, signed for Trabzonspor in 2010 after a season on loan at Eskisehirspor from Fenerbahce.
Meanwhile, Galatasaray also confirmed that they have finally reached an agreement with Kayserispor regarding the transfer of Morocco international Nordin Amrabat.
The former PSV attacker had long been linked with a move to Istanbul, but only now have the league champions come to terms with the player's employers, with reports suggesting that the 25-year-old will cost Gala €8.5 million.