The Norway international has yet to score for the Cherries this season but continues to be linked with a return to the English Premier League.
Newcastle United and West Ham United have both been credited with an interest in the 28-year-old striker, whose current deal at the Vitality Stadium is due to expire at the end of the campaign.
However, according to The Sun, the Toffees are in talks with the Cherries over a swap deal, with the Dean Court outfit keen on Toffees pair Jarrad Branthwaite and Jonjoe Kenny.
King has scored 48 goals in 161 Premier League appearances.
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