But it remains to be seen whether the Clarets reignite their interest in the 29-year-old.
Back in January, the Clarets looked into King's situation, with him then at Bournemouth and out of contract this summer, and keen to return to his native North West.
However, with his wages outweighing Burnley's top-earners, the former Manchester United youngster ended up at Goodison Park, where he signed a short-term deal.
He made 11 appearances as a substitute for Everton including one in Burnley's 2-1 win there in March, but he will be allowed to leave the Toffess when his deal expires at the end of this month.
Norwich City were keen, but they have reportedly ended their interest due to the player’s wage demands.
In January, with West Ham interested, King was believed to have asked for as much as £120,000 a week to join the Hammers, but is now believed to be asking for half that - which is still more than Norwich are prepared to pay, but is more in line with the top end of Burnley's budget, and, helpfully in terms of Sean Dyche's squad depth, is capable of playing out wide as well as down the middle.
King has also been linked with Newcastle United, where he could team up with former Cherries teammates Ryan Fraser, Matt Ritchie and Callum Wilson, and again with West Ham.