The 19-year-old has come through Fulham's academy and has already established himself in the first team, with his performances attracting attention from several major clubs.
Arsenal, Liverpool and Borussia Dortmund are all monitoring King, according to 4-4-2.ch, with the midfielder's technical ability, vision and finishing among the qualities that have caught the eye.
Arsenal manager Mikel Arteta is understood to be particularly impressed by the youngster's qualities, while Liverpool have also been following the England U21 international for some time.
Borussia Dortmund have joined the Premier League clubs in keeping track of King's development as he continues to establish himself at senior level.
King has made a particularly encouraging start to the campaign, scoring twice in five Premier League appearances.
That progress has increased his value, with Fulham understood to be seeking around €60 million for the youngster should they eventually decide to sell.
The West London club are not expected to consider a departure during the January transfer window, meaning interested clubs would have to wait until the summer before a potential deal could become realistic.
King's current contract runs until 2029, leaving Fulham in a strong negotiating position and giving the club little immediate pressure to cash in on one of their most promising young players.
A summer transfer could nevertheless become possible if a suitable offer arrives, with Fulham potentially receiving a club-record fee for the teenager.
For now, King remains focused on his development at Craven Cottage and is also gaining international experience with England's U21 side during the current international break.
His emergence as a regular Premier League player at just 19 has ensured that his situation is likely to remain one to watch ahead of next summer's transfer window.