This is according to the Daily Star, who cite ESPN as reporting United boss Louis van Gaal wants to sign at least two new defenders before the summer transfer window closes on September 1.
Van Gaal desperate for defensive reinforcements
Van Gaal has implemented a 3-4-1-2 formation at Old Trafford following his arrival earlier this summer - a system which appears to suit the Red Devils' judging by their impressive pre-season performances - and needs defensive reinforcements following the departures of top centre-back duo Rio Ferdinand and Nemanja Vidic.
Chris Smalling, Jonny Evans and Phil Jones are currently United's only first-team centre-back options, although Tyler Blackett and, to a lesser extent, Michael Keane both made a positive impression in pre-season.
Van Gaal, however, knows that recruiting at least one more top quality centre-back is crucial if the Red Devils are to push the likes of Arsenal, Liverpool, Tottenham and Everton for a top-four finish.
Benatia: Serie A's best defender
And Benatia, who is widely considered one of the best central defenders in Serie A, ticks all the boxes.
The 27-year-old enjoyed a superb season with the Giallorossi following his €13.5 million move from Udinese last summer and has been linked with many top clubs aside from Manchester United and Chelsea. Arsenal and Premier League champions Manchester City have also been linked with the north African over recent weeks.
City, though, have dropped out of the race after splashing out £32 million on FC Porto's France international Eliaquim Mangala, while Arsenal spent £16 million on Southampton youngster Calum Chambers.
United and Chelsea in direct battle
That leaves Manchester United and Chelsea looked in battle for Benatia, who Van Gaal has assigned the Red Devils' executive vice-chairman Ed Woodward the task of signing.
Van Gaal is reportedly frustrated that United are entering the 2014/15 campaign having signed just two new players - and wants several deals tied up before the transfer window closes.
Watch: Mehdi Benatia scout report
Daley Blind and Marcos Rojo
The English giants are believed to be on the brink of signing versatile Ajax defender Daley Blind, who has been spotted near Manchester this week ahead of a £15 million move, while Argentina international Marcos Rojo is currently attempting to force Sporting Lisbon to let him move to Old Trafford by refusing to train.
The acquisition of Blind, Rojo and Benatia would leave Manchester United in a significantly stronger position to challenge for a top-four finish this season - but time is running out for the 20-time English champions.