Flashscore revealed recently that leading Premier League clubs are beginning to monitor King, whose performances this season have attracted growing attention despite Fulham's difficult start to the campaign.
Barcelona have even been linked with the 19-year-old recently, underlining the potential scale of the interest developing around him.
Fulham's stance has been that King is not for sale, with the club viewing him as a major part of their future.
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However, their failure to win a league game so far has placed that position under greater scrutiny, particularly with the prospect of offers arriving for one of their most impressive performers.
King is approaching an England senior squad call-up and has continued to stand out even as Fulham have struggled for results. His individual progress has contrasted sharply with the team's problems, increasing the risk that his future could become a wider talking point if their form does not improve.
Dropping into the Championship is not an attractive prospect for a player whose ambitions are growing at such a rate.
Fulham therefore have more than their top-flight status to consider as they try to arrest their decline.
The situation also adds another layer to the debate surrounding Alvaro Arbeloa, with the club's hierarchy desperate for their managerial appointment to succeed.
Sources had previously indicated that Fulham would not seriously consider replacing Arbeloa before the second international break, reflecting their desire to give him time to turn things around.
However, the defeat to Ipswich represented a new low and has intensified the pressure ahead of crucial matches against Hull City and Coventry.
Those fixtures could prove significant in determining whether the club can maintain its commitment to Arbeloa or whether further changes will be considered.
The consequences extend beyond the manager's position. Fulham must also convince their leading best talent that their futures remain best served at the club, even if the team's current trajectory threatens to undermine their ambitions.
King's immediate future remains in Fulham's hands, and their preference is still to keep him. But as interest from major clubs grows, the prospect of relegation would make that task considerably more complicated.