The Reds beat Fulham 4-3 in the most dramatic of circumstances on Sunday, before City played out an entertaining 3-3 draw at home to Spurs, which made it three straight Premier League games in which Pep Guardiola's side have failed to win.
After 14 games played in 2023/24, defending champions City sit third – a point behind Liverpool and three behind league leaders Arsenal, who beat Wolves 2-1 on Saturday.
Speaking after the draw at the Etihad, Carragher predicted that there will be a title race this season.
The ex-Liverpool centre back told Sky Sports: "I know you [Micah Richards] don't want to hear this but, for neutrals, it's good to see City drop points. You think of them winning the treble last season and you're thinking, 'who is ever going to stop them?' You feel like you've got a title race now. You saw where Arsenal were last season and City still pegged them back. It feels like it could go to the last months of the season because of the situation City find themselves in."
Carragher conceded that he still has City – who have won five of the last six Premier League crowns – as his overall favourites, but insisted that Liverpool and Arsenal's previous experiences could help them in the second half of the season.
He continued: "If you asked us all, I think we'd still fancy Manchester City for the title. We've seen this before with City, where they go on a run during the second half of the season and win 10 or 12 games on the bounce and go away from other teams. Maybe, this season, Arsenal will be stronger from their experience in those moments and we know Liverpool have been there before."
Meanwhile, Carragher's fellow Sky pundit Roy Keane added his own doubts over City, saying: "I wouldn't bet against City but if you asked me two or three months ago, I would be City all day. Now I'm starting to be doubtful about them winning the title.
"There is a reason it hasn't been won four years in a row. I'm certainly looking at Arsenal and Liverpool."