Marca claim that Kroos has agreed personal terms with the Champions League holders, and it ready to sign a five-year contract worth €5m per year after tax (£4m).
However Real still have to strike a deal with Bayern Munich, with the midfielder expected to command a fee around €30m (£24m).
Kroos, who is currently at the World Cup with Germany, has a year left to run on his contract and would be a free agent next summer.
Despite playing an important role in Bayern's Bundesliga-winning side, he is reportedly disillusioned at not being an automatic first-team choice under Pep Guardiola.
The 24-year-old has long been linked with a move to Manchester United, whose new manager Louis van Gaal gave him his Bayern debut.
However despite rumours United were prepared to pay him a huge £260,000-a-week, Kroos dismissed claims he was ready to move to Old Trafford.
Before the World Cup he said: "I am expecting to play for Bayern Munich next season. There was a lot of speculation, but Manchester is and was not an issue."
OUR VIEW: Oh, how we long for the days when clubs at least pretended to agree a fee with each other before opening contract negotiations. Although Kroos is busy in Brazil, there's every chance his agent has been talking turkey with Real, and on the face of things he looks the perfect fit for a team looking to bolster their options in central midfield. While Kroos has distanced himself from United, Madrid naturally appears a more attractive option given the appeal of Champions League football. Though even with his contract running out, €30m seems a remarkably low price for one of Europe's leading midfielders.
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