The Metro claim Ramos has told both Madrid and United that he wants to play at Old Trafford next season amid his on-going contract dispute with Real.
The 29-year-old, who joined the La Liga giants from Sevilla in 2005, is believed to be unhappy that Real let his contract enter its final two years.
United had a £35m bid for the Spain international knocked back last week, but are understood to have returned with a revised offer worth £49.6m.
Louis van Gaal is desperate to land a world-class centre-half this summer, but Real are reportedly not prepared to sell Ramos for anything below £64m.
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Manchester United make £49.6m bid to sign Serio Ramos
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