The Blues have already secured the signings of Jordan Pickford and Davy Klaassen so far in this transfer window but have no intentions of stopping there.
Henry Onyekuru is close to signing a deal from Eupen in Belgium while Sandro Ramirez is likely to finalise his move from Malaga once the U21 European Championships are concluded.
Keane can now be added to that growing list of incoming players as Ronald Koeman looks to bolster his defensive options.
The Dutchman is believed to have made Keane, who enjoyed a superb first season in the Premier League with the Clarets last season, his first choice to improve his defence.
Talks between the two clubs have progressed over the past few weeks and a £25 million deal is now close to being agreed upon.
Keane had looked to be on his way to former club Manchester United after they showed interest in his signature but manager Jose Mourinho instead elected to bring in Benfica defender Victor Lindelof.
That left a move for Keane dead in the water and he turned his attention to Goodison Park, where he knows he will play regularly, which is vital for a player eyeing up a place in the England squad for next year’s World Cup.
Interestingly, former England international Stuart Pearce came out saying he was surprised one of the top six sides wasn't pursuing the player ahead of Everton - as far as the Blues are concerned, their loss for snoozing on the United Academy product.