The Turkish champions have been on the hunt for a right-sided defender over the past week following the sale of Sacha Boey to Bayern Munich.
After recouping €30m (£25.5m) for the highly-rated Frenchman, Gala are set to move to replace Boey ahead of today's deadline.
Fabrizio Romano claims that Gala and Forest have agreed a deal which will see the Ivorian move to Istanbul once he returns from the Africa Cup of Nations.
The report states that it will be a permanent deal, with the fee yet to be disclosed, but all that is left is for the contracts to be signed.
Aurier will be away for another few days at least after his Ivorian side knocked out holders Senegal on penalties on Monday night at AFCON.
The 31-year-old has not featured for Forest since their 1-0 defeat at home to Everton back in early December, as he entered the competition with an injury, but he has featured in three of their four games at the finals.
That looks set to be his final appearance for the club though, with a move Turkey now almost completed, ending his 18-month stay at the City Ground.
Aurier played 41 times across all competitions after joining in the summer of their return to the Premier League in 2022.
Forest do have plenty of cover in that position already though, meaning a replacement is unlikely to be bought on deadline day.
World Cup winner Gonzalo Montiel has covered since Aurier's departure to AFCON, and Neco Williams also remains at the club despite loan interest from elsewhere.
Despite playing much of the season at left-back, Ola Aina is also a right-back by trade, and can offer further strength-in-depth once he returns from Nigeria duty.
For Aurier, a move to Galatasaray will reunite him with former-Tottenham Hotspur team mate Tanguy Ndombele, and Ivorian compatriot Wilfried Zaha, but his debut will be determined on Ivory Coast's progress, as they face Mali on Saturday evening in the quarter-finals.