The club believe recent improvements under Alvaro Arbeloa will help convince their prized young talent that his future remains at Craven Cottage beyond this season.
King has become one of Fulham's most important players this season, starting every Premier League game at just 19 years old.
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His performances have attracted attention from several leading English clubs, including Chelsea, Liverpool and Arsenal, who sources say have been keeping an eye on his development.
There had been some concern over his longer-term prospects when Fulham found themselves at the bottom of the table, particularly given King's potential and the possibility that he could outgrow the club if they were to suffer relegation.
However, the mood has improved following a positive response on the pitch. Fulham are unbeaten in their last three Premier League games and there are growing signs that Arbeloa is beginning to get more out of his squad.
There is confidence within the corridors of power that Arbeloa can be given the backing to continue in his position throughout the remainder of the calendar year, taking him into the January transfer window with scope to strengthen the squad through a couple of new additions.
That is considered important not only for improving results but also for demonstrating to the club's most valuable young players that there is a positive direction of travel.
King is under contract until 2029, and Fulham are determined not to lose him. They recognise that his performances are likely to attract further attention, but believe keeping him happy and continuing to provide the right environment for his development will be crucial.
At 19, King is at an important stage of his career, and regular first-team football remains a significant part of his progression. Fulham want him to see a future at the club rather than view his next step as one that has to come elsewhere.
The expectation is that interest will continue, particularly if King maintains his current level of performance. For now, though, Fulham are calm about the situation.
Dean Jones has been a transfer news and football insider for Flashscore since 2025, as well as a journalist for leading UK newspapers and websites over the past 20 years, establishing himself as an authoritative and trusted voice in the industry.
