"After several days of careful consideration, I have decided that my footballing future lies at Goodison Park with Everton," Johnson told Everton's official website.
"I find the prospect of playing for one of the biggest names in British football impossible to resist and I am now just hopeful that I can swiftly complete my medical so that I can sign on the dotted line and commit myself to the club.
"Whilst I do believe that Everton will afford me the best opportunity to fulfil my twin ambitions of winning major honours and again representing England at senior level, the last few days have been both hectic and demanding.
"I would like to thank both Wigan Athletic and Bolton Wanderers, firstly for having sufficient faith in my abilities to lodge a bid and secondly for being so courteous and professional during the course of our negotiations. I was very flattered by the interest of both clubs.
"I would also like to thank everyone at Crystal Palace - especially Iain Dowie and the fans - for their support and dedication during my time at Selhurst Park. In many ways I am sorry to be leaving but I wish them well for the future."
Meanwhile, Bolton's hopes of signing South African midfielder Quinton Fortune have been hit by the news the player favours a move to Celtic.
He told the Sunday Mail: "Keano phoned me up - we were friends at United. I was in Glasgow on Friday to see things for myself and it was terrific.
"Celtic are one of the greatest clubs in the world and the history and passion around the place is unbelievable.
"If you asked any player in the world to play there, they'd do it.
"I don't want to make too much of a comment or be disrespectful to Celtic because I haven't had a medical or signed anything.
"But, from what I saw of them on Friday, it would be an honour to play there.
"There are a few clubs involved, but I can't say too much at the moment because I don't want to come across as being cocky or disrespectful to any of them."