United finished second last season and have added Jadon Sancho as well as Raphael Varane to their squad this summer.
In a debate with Gary Neville on Monday Night Football, Carragher claimed manager Ole Gunnar Solskjaer should be challenging for the title with his current squad.
Neville kicked it off by claiming: "I'm struggling to understand why United aren't going that extra step in this transfer window.
"Sancho was £70 million but that was last year's money. In regards to United, they've always gone after the best English or British player in the Premier League, historically.
"Why is [Anthony] Martial, [Dan] James, [Jesse] Lingard, you're talking about offloading wages there and potentially getting a transfer fee in.
"If Guardiola, Klopp or Tuchel had #MUFC's squad, you'd feel they could win the league."@Carra23 and @GNev2 discuss Manchester United's spending under Ole Gunnar Solskjaer and whether he is under pressure to deliver this season...- Sky Sports Premier League (@SkySportsPL) August 23, 2021
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"Why aren't they going in for Harry Kane this week? I think if Kane went to United, they could get up to 90-95 points.
"They still need to replace [Edinson] Cavani and Martial next summer. Martial isn't going to do it, Cavani only has one year left.
"They're going to have buy a centre-forward at some point, why not get the man now and give Ole the impetus he needs to get that points total.
"I don't think he can with just Sancho and Varane."
Carragher then hit back by claiming the other three top four managers would be able to challenge for major honours with the Red Devils right now.
He responded: "You're talking about Chelsea going after [Romelu] Lukaku, he scored on his debut and all of a sudden it's 'Chelsea are going to win the league'.
"United finished above them last season, they bough Varane and Sancho, I don't get this feeling of 'United can't compete'.
"If Guardiola, Klopp or even Tuchel had United's squad, I feel they could win the league, I really believe that.
"United have spent the money City have done, spent the money Chelsea have done, so this thing of building, yes.
"But I don't think United not winning the league this year. Finishing third or fourth, that can't be acceptable.
"My point is with the squad that is there, you never want to say United could win the league but that squad there is £50m-plus in almost every position.
"What's lacking at United to really for the league? Is it the team, the squad or the manager?"
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