The former Manchester United product put pen to paper on a five-year deal on Monday evening for an initial fee of £25m, which could rise to a record-equalling £30m - the amount paid to sign Jordan Pickford from Sunderland last month.
Keane is understood to have chosen the Toffees over a number of other potential suitors, and the 24-year-old is confident that he has made the right choice after meeting face-to-face with European Cup-winning defender Koeman.
"I'm absolutely delighted to be at Everton, a fantastic football club with passionate fans and a great tradition and I can't wait to come into training tomorrow to meet all the lads," he told his new club's official website. "First and foremost, the manager was a big factor in me coming to the club. He played in my position when he was a world-class player for a great Barcelona side and a great Dutch side and the style of football he likes to play will suit me, I believe.
"I watched a lot of Everton last season and I believe I really suit how the team likes to play. I feel like this is a great place to come and continue my development as a player. He's convinced me that he can still improve me in certain areas of my game. I'm only 24 which, for a centre-half in the Premier League, is relatively young. I've still got a long way to go and I know this and hopefully under his guidance I'll keep improving.
"You always want to play for a manager who believes in you. I wanted to come somewhere I was wanted by the staff, players and fans as well. I feel like Everton is the perfect match. I've also got to say thanks to the chairman Bill Kenwright - he's been absolutely fantastic getting the deal through.
"I also spoke with Steve Walsh, who I knew from my time at Leicester four or five years ago, and they really convinced me that this was the place to come and that Everton is the perfect club for me."