The midfielder has been one of the stars of the 2018 World Cup so far after helping Russia confirm their place in the knockout rounds.
Golovin has grabbed a goal and picked up two assists in the tournament and he was reportedly being watched by Arsene Wenger last season.
The 22-year-old is still on Arsenal's radar and Unai Emery has reportedly given the green light for the club to make their approach.
Reports claim he will cost £26million and he could soon join new arrivals Stephan Lichsteiner and Bernd Leno at the Emirates.
Arsenal are also determined to snap up Sampdoria midfielder Lucas Torreira and Sokratis Papastathopoulos of Borussia Dortmund in the coming weeks.