The 19-year-old has made major strides as a first team player at Craven Cottage in the last 12 months after racking up 36 appearances across all competitions last season.
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The England U21 international is under contract with Alvaro Arbeloa's team until 2029 and any sale is expected to reach around €60M from Barcelona's perspective.
Several clubs are monitoring King's progress with real interest, and as per the latest from TEAMtalk, Barca could be rivalled by Chelsea, Liverpool, Manchester United, Borussia Dortmund, RB Leipzig, Inter Milan and Atletico Madrid.
A January offer is unlikely, as Barca and others consider their options, but King's close control and ability to break forward from midfield has impressed scouts from across Europe.