The 23-year-old is into the final 12 months of his contract at Goodison Park and has rejected the offer of a new deal amid reported interest from Spurs, Chelsea and Arsenal.
Former Toffees captain Neville believes that it is time for Barkley to leave the club, with Spurs the ideal destination for the England international.
"I think he's a top, top player and he needs to play for one of the clubs playing in the Champions League," Neville told The Debate.
"If you look at the clubs to go to, managers to play for, I can only see one - Mauricio Pochettino.
"For Ross Barkley you have to sign for the right manager, somebody that's going to improve him, and more importantly coach him and make him a better footballer. I think Pochettino is that man."
Barkley did not take part in any of Everton's pre-season schedule, partly due to a groin injury.