The 21-year-old has attracted attention from several major European clubs following his arrival from Lokomotiv Moscow, with his performances in the Champions League further strengthening his reputation.
According to reports in Turkey, Barcelona and PSG are among the clubs keeping track of his development, although neither has made a formal offer.
Barcelona's interest in Batrakov dates back to his time at Lokomotiv, where he established himself as one of Russia's most promising attacking midfielders.
His performances attracted considerable attention before Galatasaray secured his signature during the summer transfer window.
The Istanbul club paid €25 million to Lokomotiv for the midfielder, who signed a five-year contract running until June 2031. The agreement also includes a clause entitling the Russian side to 15 per cent of any profit Galatasaray makes from a future sale.
Batrakov has already demonstrated his ability against Barcelona this season, scoring the opening goal when the Spanish side visited Galatasaray in the Champions League.
Barcelona eventually recovered from that setback to secure a 4-1 victory, but the midfielder's contribution provided another opportunity for the Catalan club to assess a player they had previously monitored.
His early performances in Istanbul have been encouraging. Batrakov has registered two goals and two assists across his first nine official appearances for Galatasaray, quickly establishing himself as an important part of the team's attacking plans.
PSG's interest is also understood to predate his move to Turkey, with the French club having followed his development during his time in Russia.
Galatasaray are unlikely to consider selling a player they only recruited recently, particularly given the length of his contract and the investment made to secure his services.
The Turkish club also have an incentive to demand a significant transfer fee should interest from Europe's leading teams develop into concrete offers.
Batrakov's technical quality, creativity and ability to contribute goals make him an attractive prospect, but Barcelona and PSG will have to assess whether a deal is financially realistic.
For now, both clubs are monitoring his progress rather than pursuing an immediate transfer.