Spurs strengthened their wide options over the summer with the additions of Brennan Johnson and Manor Solomon, while Dejan Kulusevski converted his 18-month loan deal into a permanent move.
However, the North London club are seemingly keen to recruit another winger in the January transfer window following a couple of injury setbacks.
Solomon is out of action for the next few months with a knee injury, and he is joined on the sidelines by long-term absentee Ivan Perisic, who suffered an anterior cruciate ligament injury in September.
With Spurs keen to bolster Ange Postecoglou's wide options, a recent report claimed that Tottenham had sent scouts to watch Akturkoglu in action.
According to Turkish outlet Aksam, Tottenham have made contact with Galatasaray over a potential transfer, as they take the first steps to securing Akturkoglu's signature.
The report claims that Spurs have asked Galatasaray for information about a possible deal and are keen to hold conversations with the player's representatives.
Arkturkoglu has also been attracting interest from several other clubs after scoring five goals and seven assists in 16 competitive appearances this season.
Tottenham's North London rivals Arsenal have been linked with the Turkey international, while Manchester United and Liverpool are believed to be among his admirers.
With Spurs expected to face competition for Akturkoglu's signature, they will look to make use of the relationships built with Galatasaray over the summer.
Tottenham chairman Daniel Levy held negotiations with Galatasaray over two deals, which saw Davinson Sanchez join Okan Buruk's side on a permanent basis and Tanguy Ndombele complete a loan switch to the Turkish champions.
With Galatasaray determined to retain Akturkoglu, they are keen to tie him down to a new contract, with his current deal set to expire in June 2026.
Spurs are also said to be monitoring the progress of Akturkoglu's teammate Victor Nelsson, who operates as a central defender.