The 34-year-old is out of contract at the end of this season but remains in negotiation with Real over an extension.
Calciomercato reports United are hoping Ramos will consider their own contract offer.
The Red Devils would be prepared to match Real's wage offer on a two-year contract.
Former United boss Louis van Gaal unsuccessfully attempted to sign Ramos in 2015.
United are also reportedly ahead of Manchester City in their pursuit of Spain's record cap holder.