Broadcaster Skysports reported that Keane, 24, underwent a medical at Everton's Finch Farm training centre on Monday following three successful seasons at Burnley, who he joined from boyhood club Manchester United.
Everton broke their transfer record last month by signing England Under 21 goalkeeper Jordan Pickford from Sunderland in a deal that could rise to £30 million and the Keane transfer equals that.
"He's 24 years old, his best years are to come, he's an England worldwide, too, and I couldn't be more pleased that he's now a Blue", he added. I wanted to come somewhere I was wanted by the staff, players and fans as well.
The ex-Manchester United ace added: "He's convinced me that he can still improve me in certain areas of my game".
"First and foremost, the manager was a big factor in me coming to the club".
Everton boss Ronald Koeman has prioritised the signing of the 24-year-old this summer as he looks to strengthen his defensive options. "I know him from my days at Leicester, so I felt really wanted by everyone at the club and I just can't wait to get going now".
He also earned a call-up to England national team in March on the back of his impressive performances for the Clarets.
Koeman, 54, has invested heavily since guiding the club into seventh place in the Premier League last season.
Keane made his full global debut in a friendly against Germany in March and played in a World Cup qualifier victory over Lithuania three days later.
Only hours earlier Everton, who have also signed midfielder Davy Klaassen for £23.6 million from Ajax and Nigeria striker Henry Onyekuru for £6.8 million from KAS Eupen, confirmed the £5.2 million signing of striker Sandro Ramirez from Malaga.