The People says United are favourites to sign Benzema after Real Madrid ruled out a move for the French striker.
Sir Alex Ferguson is keen to bolster his forward line as his club face a fixture pile-up in the new year.
Ferguson has been tracking the Lyon striker for a long time and was close to signing him before Dimitar Berbatov's arrival in September.
Benzema, 20, regarded as one of the best young strikers in the world, revealed earlier in the week that it would be his dream to sign for Real Madrid, but is now keeping his options open.
Benzema said: "I like AC Milan and Manchester - I love the English, Italian and Spanish leagues."