Ole Gunnar Solskjaer's side have arguably shown title-winning form since the restart, winning four Premier League matches in a row, also extending their unbeaten run to 17 games with a 3-0 win against Aston Villa on Thursday night.
United's attack has been particularly potent of late, with the acquisition of Bruno Fernandes - alongside the rise to prominence of Mason Greenwood - inspiring the Red Devils to 11 goals in their last three games.
Gary Neville has insisted that Liverpool's front three of Sadio Mane, Mohamed Salah and Roberto Firmino remain ahead of United's Martial, Greenwood and Marcus Rashford, though says the latter have the "talent and ability" to compete.
While Carragher acknowledges United's improvement, he thinks they must bring in another world-class attacker if they are to challenge for a title.
“Liverpool possess an attacking trio that is difficult to improve," he wrote in The Telegraph. "Can we say the same of United?
“I think they need another world-class player at the top end of the pitch.
“Martial’s goal record has improved but I am yet to be convinced he is the man to fire them to a title challenge.”
Standard Sport understands Borussia Dortmund star Jadon Sancho remains a priority signing for United this summer.
The 20-year-old both scored and assisted 17 goals in the Bundesliga this season, an improvement on Martial's 15 goals and four assists thus far.
A number of reports have claimed the England international has already agreed personal terms with United.
And Dortmund sporting director Michael Zorc also hinted a move may be on the cards, telling Kicker: "If Jadon were to leave, we would once again be active in strengthening our attack."