The newly crowned Premier League champions are long-term admirers of the 25-year-old France international, and came close to signing him when he left Lyon in a £35million deal in 2009.
Red Devils boss Sir Alex Ferguson will advise his successor David Moyes - who takes over from the 71-year-old Scot from July 1 - to do everything possible to bring Benzema to Old Trafford.
Benzema still has two years to run on his existing contract at the Santiago Bernabeu, but like a number of other Real Madrid players, there are question marks surrounding his long-term future in Spain, given the fact that Jose Mourinho is poised to quit the capital club at the end of the season.
Earlier this month, Benzema's sgent, Karim Djaziri, quashed speculation suggesting his client will be looking to move, saying that Mourinho's predicament will have no impact on the player's future.
"I've said it before, and I will repeat it: Benzema is not leaving Madrid," he told CalcioMercato.
"The possible departure of Jose Mourinho does not change the situation. A transfer is not an option."
However, fresh reports are now circulating that Manchester United will renew interest in Benzema as soon as the transfer window reopens, and look add him to the club's fearsome forward line ahead of the 2013/14 campaign.