Dutch giants Ajax are believed to have already failed in an attempt to sign the striker.
Current Dutch champions Feyenoord are looking to promote Zirkzee to their U19 team next season, but he is understood to be sirously considering a move to England.
The hotshot has scored once in four games for his country at U16 level.
Though there is interest from a number of clubs, it looks like Arsenal are currently leading the race to sign the 16-year-old.