According to 'Sky Sports' the teenage striker will sign a five-year-deal with Madrid from Swedish side AIK Solna this week.
The youngster has been likened to the Manchester United striker due to his physical playing style and his nationality.
However, the transfer cannot be made official until the summer as Real Madrid are currently banned from signing players, having broken rules regarding international youth transfers.
Isak has already made 29 senior appearances for AIK Solna, scoring 13 goals. He also has two caps for Sweden and recently scored his first international goal in Sweden's 6-0 rout over Slovakia.