So, to improve over Everton, he needs to go to one of last season's top six. I highly doubt that club would be Liverpool and I think United would target a more defensive midfielder player firstly. He's not starting over Alli at Spurs and I don't think both will play together - it would be like Gerrard & Lampard for England all over again IMO.
Arsenal seem well stocked in that kind of area, although if they were to sell Oxlade-Chamberlain they'd probably want to bring in another more attacking, English midfield player and it would still look like a 'marquee' signing of sorts for Arsenal. Would he really be first choice though? Man City need midfield players and could also do with adding some more British influence, but again I'd question if he would be starting every week? Almost certainly not if Pep persists with the 4141. However, he could probably get more money to sit on the bench more often by going to City.
That leaves Chelsea, which is possible, but it's going to be a lot of rotation between Kante, Bakayoko, Fabregas and Barkley - so, again, guaranteed less playing time.
Point is, Barkley is a squad player at any of these teams and would probably warrant a £35-40m price tag for the above teams. Is it worth it?