The 18-year-old has been tipped for a big future and he is currently impressing for Norwegian side Odd.
The forward has struck five goals in 27 games this season and has caught the eye of Everton scouts.
Now, according to Norwegian newspaper Varden, the Toffees have stepped up their interest and are close to clinching his signature.
Zekhnini is reportedly flying to England for talks with the Merseysiders ahead of a permanent move.
Odd have previously rejected several approaches from various clubs for the youngster, but they have now decided to grant his wish and let him leave.