Former United and Ireland skipper Keane was impressed by what he saw from Liverpool and City in the Community Shield match at Leicester, as Darwin Nunez came off the bench to swing the game as the Reds' ran out 3-1 winners.
And Keane suggested the big two will not be challenged when the season gets underway next weekend.
"Liverpool looked a bit sharper, were a bit more direct, they created the better chances and were the better team, but there is a week to go yet," he told ITV Sport.
"City looked slow out of the blocks last year but it's about how you finish, and City are good at that.
"I look around the Premier League and Tottenham are the one team that will think they can get in the mix this season, but Liverpool and City are a class apart.
"We saw what they are all about again in this game. The passing and movement was top class and even though Klopp said they weren't at their best, I thought this was a great game between two outstanding teams.
"They will be one and two in the Premier League again this season and it's just a case of what order they will be in.
"Spurs might be closest to them this year, but Manchester United? I just don't see it. I hope they prove me wrong, but they don't look good enough for me.
"This has been a good day for Liverpool. A new striker, good habits to get into winning, other teams might be sitting watching this thinking are these teams going to ease off?
"Not with Pep and [Klopp in charge, the hunger and desire from both teams. Klopp got a big smile, looks refreshed, got a new contract - good luck stopping these teams."
City striker Erling Haaland struggled in his first big test after his summer move from Borussia Dortmund, but Keane suggested it was not all gloom for the Norwegian forward who fluffed his lines with a glorious chance in the closing seconds of the game.
"I'd be encouraged from what I saw from Haaland today," added Keane. "Okay he missed two chances but he's getting chances and it's nice to see with the big hype around him - it'll humble him a little bit and it's going to be hard work.
"There's no doubt in my mind he's going to score goals for fun, his movement, chances he had - he'll be disappointed."