Everton have spent heavily this summer, and there is a feeling of great optimism around Goodison Park ahead of the new campaign.
And Neville believes that if Liverpool sell Philippe Coutinho, Everton may have a chance of catching their Merseyside rivals.
“Look, I’ve never seen a more positive set of supporters,” Neville said, when asked if Everton could catch Liverpool. “I think Everton can make the jump into the top six. I think this is their season to make that jump.
“If Liverpool get rid of Coutinho. Liverpool and Chelsea have finished outside the top six in recent years. They’ll be looking over their shoulders at Everton.”
Craig Bellamy, who was asked the same question, was more sceptical over Everton’s chances of catching Liverpool.
The former Reds striker admitted that the Toffees should be aiming to crack the top six this season, but he feels that is a step too far currently.
“Everton should be looking to break into the top six, but I still can’t see them getting it,” Bellamy said.
Everton will be looking to send out a statement in their first match of the season, which comes against Stoke City on Saturday.
If the Toffees can win the fixture then the momentum collected through their summer business is only set to continue to build.
Liverpool, meanwhile, face Watford at Vicarage Road in their Premier League opener.