Walsh has been credited with helping discover bargain signings such as winger Riyad Mahrez and Kante for the Foxes, who last season became top-flight champions for the first time in their 132-year history.
Professional Footballers’ Association player of the year Mahrez joined from Le Havre in 2014 for £400,000, while Kante was signed from French club Caen for about £6m.
“I am genuinely excited to be here at Everton and very much looking forward to getting down to work,” said Walsh.
“I know the club well, my roots are in the north west and my brother, Mickey, was an Everton player in the Bob Latchford era and always proud of that fact.”
Dutchman Koeman added: “It is really good for the club to have someone in Steve who has shown his quality in his job at a number of clubs over many years.
“That is the experience and quality he will bring to us at a good time in the season.”
Leicester’s director of football Jon Rudkin praised Walsh’s “signification contribution to the most successful period in the club’s history” and said he had accepted the decision “with a heavy heart”.
Pat Murphy, BBC Radio 5 live:
“Arguably, Walsh became the most important figure in Leicester’s football department because he kept delivering unpolished gems from his countless scouting missions without breaking the bank.
“Although he signed a new and improved contract only in May, he has been tempted away by what he sees as a career promotion and the vision of Everton’s majority shareholder Farhad Moshiri.
“The London-based Iranian businessman has ambitious plans for Everton, with money seemingly not an issue. Landing first Koeman as manager and now Walsh within a month is a double coup.
“Leicester will be seeking full compensation for the loss of Walsh but just as in the pursuit of Koeman, money is no longer a thorny problem at Everton.”