Ramos has been offered a new deal at the Bernabeu, with his current contract set to expire in 2017.
The 28-year-old has been linked with the likes of Arsenal and Chelsea, but United are also keen.
And the Red Devils will apparently try to lure Ramos to Old Trafford with the offer of £105,000-a-week after-tax.
Ramos has not publicly indicated he wants to leave Real, having been at the club since his £21million move from Sevilla in 2005.
The La Liga giants are unlikely to sanction his sale, given his importance to Carlo Ancelotti.
The Spanish international has won every major trophy at club and country level with Real and Spain.
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