Bruma has been attracting wide-spread interest from the Premier League after a brilliant start to the season.
Galatasaray and Besiktas are both locked on 10 points in the Super Lig and the winner of the derby could end up moving into pole position.
The Gunners are set to keep tabs on the Lions rising star in the derby to monitor how he copes with his biggest game of the season thus far.
Jose Mourinho ordered his scouts to keep tabs on Bruma ever since the two sides faced each other in a pre-season friendly over the summer.
The young winger put on yet another outstanding display scoring in the 5-2 defeat against United.
“He congratulated me for the performance and said he was impressed with me,” Bruma was quoted as saying by Turkish-football.com. “He promised to closely monitor my work and said I could play for the national team soon.
“Of course it was amazing to hear this from Mourinho and he said he will keep an eye on how I do this season.”
The Portuguese winger was destined for greatness when he burst onto the scene as a youngster as Sporting Lisbon but he never quite developed as expected after being held back by a series of injuries.
Bruma has made a full recovery and is still a youngster with a bright future ahead of him.
The 21-year-old has a goal and two assists in the league for the Lions and has been awarded man of the match on two occasions.
Galatasaray are currently in talks over extending his contract to 2020 and adding a €40 million release clause into the deal.
In other Man Utd transfer news the Daily Star reported today that head coach Jose Mourinho blocked Michael Carrick leaving the club over the summer.
The experienced midfielder was linked with a move to Newcastle United but Mourinho wanted him to remain at the club.