The Liverpool legend and Sky Sports pundit told The Last Word with Stan Collymore podcast (h/t Dominic Booth of the Manchester Evening News) he thinks the Red Devils are closer to Premier League success than many believe.
Carragher explained he is impressed with the way United have restructured this summer after the sale of Romelu Lukaku to Inter Milan and the recruitment of Harry Maguire, Aaron Wan-Bisakka and Daniel James.
"I know from being at Liverpool when we've had times outside the top four as a player, I'd always say it's never as bad as you think.
"Sometimes you just get one player or a couple of players who leave, and something just lights up. I'm not saying that takes you to the title, but I don't think Manchester United are miles away, I really don't.
"In some ways I admire what they've done and the signings they've made. Could they have replaced Lukaku? I think it's about time (Marcus) Rashford was given a go through the middle."
Ole Gunnar Solskjaer's team began the Premier League season in style with a 4-0 win over Chelsea at Old Trafford on Sunday. Rashford scored a brace, with United playing a high-intensity game against their rivals for UEFA Champions League qualification.
Maguire made his debut for United in the victory and produced an excellent display at the heart of the defence. The £80 million fee paid to Leicester City for the England international was a world record for a defender. The 26-year-old is expected to provide the long-term foundation of Solskjaer's starting XI.
Carragher added he believes Maguire was one of the best options for United during the transfer window this summer.
"The defenders they brought in in Maguire… I think it's a good signing. People are questioning the price tag but he had a five-year deal and they (Leicester) knew (United) were desperate. Is there anyone better out there you could get? Probably not. People saying (Matthijs) de Ligt but he's going to sign for Juventus before United. Could they have brought one or two more in? They've still spent more than anyone. I don't think United are miles away the way people talk."
The immediate future of Paul Pogba will be resolved in the coming weeks, with European transfer windows set to close by September 2. The retention of the Frenchman is key for this season. The midfielder grabbed two assists against Chelsea, and United have no immediate replacement in their squad to compensate for his talent.
A top-four finish in the Premier League, supported by extended runs in cup competitions, would constitute success this term. United are not expected to challenge Manchester City or Liverpool at the top of the league during the next 12 months, but a strong season will see them well-positioned heading into next summer's transfer window.