Fulham have been favourites to sign Johnson in recent weeks, despite interest from Premier League rivals Sunderland and Wigan Athletic.
The former Crystal Palace forward is thought to favour a return to London for family reasons and could become a Fulham player in the next 24 hours.
He had been left out of Everton's squad for the pre-season trip to the United States and the Toffees have now revealed a deal has been agreed with Fulham.
A statement on the club's official website read: "Everton can confirm that it has agreed an eight-figure sum with Fulham for the services of Andrew Johnson.
"This is subject to the player agreeing personal terms and negotiating a medical."