Arsenal won three Premier League titles in the space of seven years at the turn of the century but have not come close to lifting the title in recent seasons.
The north London club have struggled since legendary manager Arsene Wenger left, finishing just eighth in the Premier League in the last two seasons.
This season, Arsenal have made their worst-ever start to a Premier League campaign, losing their first three matches without scoring a goal under Mikel Arteta.
Arsenal are traditionally considered one of the biggest teams in England but Neville believes a number of sides have now surpassed the Gunners, who he no longer views as a top-six club.
Asked to name the top-six in England, Neville told LADbible TV : 'Liverpool, Manchester United, Manchester City, Chelsea and then Tottenham and Leicester based on recent seasons.
'Arsenal fans will probably come up with some stat now… they think I hate them anyway! They drain me! Start winning… I want to start praising them again!'
Earlier this week, Arsenal legend David Seaman claimed the players at the Emirates were 'getting away with murder' under Arteta, who replaced Unai Emery as manager in 2019.
'The players haven't turned up, even towards the end of last season,' Seaman told talkSPORT .
'I know Arteta was getting a lot of stick but the players weren't turning up and I was like, "hang on, these guys are getting away with murder", because there's not a lot being said about them and they're not performing to their best.
'They've got to turn up this weekend, it's a massive game and I fear the worst if we get beat, because the fans will turn. They're not happy with two seasons of finishing eighth, it's nowhere near good enough.
'We're at that point now where the board have chucked some money at the team and the squad,' said the former goalkeeper.
'They've put their money where their mouth is and it's time for the players to do the same.'
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