King joined Bournemouth in 2015 and scored 53 goals in 184 games, but his contract was due to expire at the end of the 2020-21 season.
Fulham were also interested in the Norway forward but King opted for the Toffees, who have paid a nominal fee.
The deal is believed to be a six-month contract with an option to extend it at the end of the campaign.
King was linked with a move to his first club Manchester United in the January 2020 transfer window but Old Trafford boss Ole Gunnar Solskjaer signed Nigeria forward Odion Ighalo instead.
In 2016-17 Bournemouth finished ninth in the Premier League – their highest position – and King’s 16 league goals were a key part of that successful season.
He netted six league goals last season as Bournemouth were relegated to the Championship.
They are sixth in the Championship this season but King has failed to find the net in 12 league matches – although he scored three times in the FA Cup.