However, Barton is not convinced by De Boer and the boyhood Everton supporter has stated that the Toffees should instead try to attract former Chelsea manager and current Manchester United target Mourinho.
He told talkSPORT: "You need tried and tested in the Premier League. No disrespect to De Boer, he is a good manager, and will prove that in time, but I don't think he's for Everton now.
"If you could get Koeman, I think he's proven himself in the Premier League. If you look at his track record as a player and as a coach, he has a track record of winning.
"If you can get Mourinho, you've got to get Mourinho. He has to be first choice, just because he's as good as anyone. You've got to forget what's happened this year at Chelsea, if Mourinho is given the right money and assurances he's one of the best out there.
"Failing that [hiring Mourinho] I would go Koeman, and if all else fails, take a chance on De Boer. But if you take a chance on De Boer you could be sitting very close to a relegation battle next season. It's not guaranteed that you'll get an upturn in form."