The 24-year-old, who has penned a five-year contract, is Everton's fifth signing of a close season which has seen the club's Dutch manager Ronald Koeman invest heavily in improving the side.
The fee represents a record sale for Burnley, who signed the central defender from Manchester United in January 2015 for an undisclosed fee put at two million pounds by British media.
In March, Keane made his full England debut in a friendly against Germany before playing in a World Cup qualifier versus Lithuania.
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($1 = 0.7729 pounds)
(Reporting by Simon Evans; Editing by Ken Ferris)
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