The 15-year-old English midfielder has been training with the Fulham first team during the international break.
The London Evening Standard reports that the likes of Barcelona, Borussia Dortmund, Manchester City and Chelsea are keeping tabs on the youngster, who does not turn 16 until April.
Fulham hope they can persuade Elliott to stay at the club, just like Ryan Sessegnon did.
Sessegnon was also courted by the top clubs in Europe when he was a schoolboy, but decided to sign a deal with Fulham and helped the club win promotion to the Premier League last season.
