Kone, 17, made the bench of Toro's game with Lazio for the first time on Monday, though coach Sinisa Mihajlovic chose not to field him.
However, TMW says scouts from United and Ajax were in the stands with the purpose to see the Ivorian attacker in action.
Both clubs are eager to ferry to Kone away from Italy, with Kone yet to commit to a pro contract. The youngster plays as a No10 for Toro's primavera team.
The United rep, meanwhile, was also in attendance on Monday to check on Lazio midfielder Sergey Milinkovic-Savic.