Falcao has already agreed personal terms with Real and talks over a fee have opened between the two clubs.
Sport says super agent Jorge Mendes is driving the negotiations, with his Gestifute management company handling the affairs of both Falcao and Di Maria.
Mendes has also raised the prospect of Real taking Falcao's Monaco teammate James Rodriguez in a stunning three-player mega deal. The Colombia World Cup star, another on Mendes' books, would move to Madrid as a direct replacement for Di Maria.
Rodriguez would love to switch to Real after the World Cup, but Monaco will need to be convinced. Having paid €45 million for Rodriguez and €60 million for Falcao last summer, the French club are insisting Real get close to their outlay for any such move to happen.
For the big stories and the banter follow us on Twitter: @tribalfootball