Gibbs, 27, has failed to feature for Arsenal so far this season and only made 11 Premier League appearances last term.
Arsene Wenger is keen to get a host of players off his books this summer and Gibbs appears to be surplus to requirements at the Emirates.
The England international has been heavily linked with a move away from Arsenal as a result, with Newcastle and West Brom expressing an interest in him.
However, Turkish outlet Fanatik claim Galatasaray are on the verge of securing his signature.
It is said Gibbs’ father, with Mesut Ozil’s agent Dr Ekan Sogut acting as the middle man, has held talks with the Turkish giants in Istanbul over a possible transfer.
Negotiations are apparently now at an advanced stage with further talks being held today.
Fanatik say Gibbs is expected in Turkey this week where he will sign his contract in a deal worth around £3.7million.
The left-back has been at Arsenal since 2004 and was promoted to the first team in 2008.
He has made 137 appearances during his time in north London and scored two goals.
But with Gibbs now entering last 12 months of his current contract it appears as if his time at the club has come to an end.
Arsenal will be looking to make up for their 1-0 defeat to Stoke on Sunday when they take on rivals Liverpool at Anfield (4:00pm).