The midfielder has enjoyed a superb campaign and has scored 15 goals to fire Sven Goran Eriksson's side to the brink of play-off contention.
David Moyes has enjoyed great success in recent seasons delving into the Championship most notably with Tim Cahill, Joleon Lescott and Phil Jagielka.
Superb season: Andy King in action against Bristol City last weekend
Leicester do not have to sell any of their squad after being taken over by Thai investors but Everton are well placed to steal a march on other clubs who have been alerted to the 22-year-old's progress.
Everton striker Ayegbeni Yakubu was loaned to Leicester in January with a view to a permanent move and he could feasibly be included in any negotiations for King.
Everton have completed the signing of young Swedish centre-back Johan Hammar, 17, on a permanent contract. The former Malmo defender moved to England last year but was ineligible to sign a contract until his 17th birthday earlier this week.
Meanwhile, Everton are also monitoring ?7m-rated PSV Eindhoven winger Balazs Dzsudzsak, 24, but face competition from Arsenal.
?Injured Everton star Cahill a major doubt for Sunderland visit Moyes brings belief and hopes FA Cup win can revitalise Everton's fortunes
?Explore more:People: Phil Jagielka, Tim Cahill, Joleon Lescott, David Moyes, Sven Goran Eriksson Places: United Kingdom
Source: Daily Mail
Source: Daily Mail