The Danish midfielder is set to leave the club as his contract at the North London side will expire at the end of this season.
And with Los Blancos hopeful of competing for all trophies this year, a move for the Dane in January could be likely to boost a midfield that is already light on numbers without the numerous injuries the Madrid squad have suffered already this season.
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A move never materialised though, and Spurs chairman Daniel Levy would have to sell him in the upcoming transfer window in order to avoid losing him as a free agent.
One potential stumbling block, though, is that Eriksen would prefer a summer move as he could then request a higher signing-on bonus as there would be no transfer fee.
Having received the proposal, Los Blancos are now considering it and if Eriksen is available for that price, they believe it's best not to miss the opportunity, especially when Levy slapped down a €250m price tag when Real Madrid first approached him for the Dane. With both Real Madrid and Manchester United interested in his signature, another European heavyweight has entered the race to get Eriksen on board.