Liverpool top the Premier League after beating Wolves 2-1 on a weekend that Man City dropped points by drawing at Newcastle.
Arsenal, meanwhile, are level with the champions, both a point behind the leaders.
Slot, in his first season at Anfield, has won five of his opening six league games, the loss coming at home to Nottingham Forest.
But Neville is not convinced that Liverpool can kick on from their third-place finish last season under Jurgen Klopp.
"They're going to have a good season it looks like and maybe get Champions League football," he said on the Gary Neville Podcast.
"I didn't feel like I was watching title winners [at Wolves] being honest with you. They're well below Arsenal and City but I don't think that's shocking anybody."
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Arteta Pep
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Neville expects Arsenal and Man City to be the ones fighting for the title come the end of the season and their rivalry is already hotting up.
Arsenal have fallen short the past two seasons when challenging City, but Neville thinks they are getting closer to ending the champions' four-year hold on the title.
Arsenal 4-2 Leicester | Premier League highlights
FREE TO WATCH: Highlights from Arsenal's match against Leicester in the Premier League
Asked about the title race, Neville added: "It was good last year, I think it's going to be good this year. I think City know that there's someone creeping up on their shoulder.
"We knew it when City were chasing us all those years ago when I was sort of retiring.
"You knew that they were getting closer every single year with the signings, with their stability and their consistency and their confidence getting better.
"There's no doubt that Arsenal are on their shoulder."
Guardiola has already seen off Klopp as a title rival, Liverpool winning just one of the last seven titles, and will be relishing this latest battle with Arteta, according to Neville.
Newcastle, Manchester City
FREE TO WATCH: Highlights from Newcastle's draw against Manchester City in the Premier League
"Pep Guardiola is smelling and feeling the challenge. He wants this challenge. He loves to overcome this type of challenge," added Neville.
"They don't want to give in City, they don't want to make it easy for them.
"But I like the way in which it is going to pan out. I think we're going to have a really good title race this season."
Slot: We have a lot to prove
Liverpool boss Arne Slot told Sky Sports after the win at Wolves that it is too early to consider his team title contenders.
"Pre-season helped. The players started to believe. We have a lot to prove when we come across the top sides. It's good that we are where we are at the moment," he said.
"I always try to be realistic, Jurgen's work has been fantastic. We have to prove we can do it in the Champions League and Premier League."
Analysis: Slot not getting carried away
Sky Sports' Adam Bate:
As Arne Slot pointed out after Liverpool's 2-1 win over Wolves at Molineux, it did not feel like a top-against-bottom game and he is reluctant to get carried away just six games in. The Dutchman insists that we should only judge when everyone has played each other.
But for now the table shows Liverpool ahead of Premier League champions Manchester City, ahead of Premier League favourites Arsenal. Five wins from six games with the best goal difference, as well as two wins from two in the other competitions as well.
There is depth in attack, unexpected synergy in midfield and the defence looks strong. Slot still wants more patience from his players but for now there are promising signs that they can add that composure without losing the cut and thrust of the old Liverpool.
Even if any fledgling title push fails to materialise - they faded last season under Jurgen Klopp - Slot is right to point out the obvious when he says that if Liverpool had started poorly, he would have been questioned. As it is, the table shows his side at the very top.
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