The Croatian midfielder’s loan deal at Stamford Bridge had expired on 30 June and once this tie to the West London club ends, they could not sign him due to their transfer ban on players not currently registered to the club.
The Blues are not allowed to register new players for the next two windows under the FIFA-imposed ban, but have a loophole with the Croatian, who was already registered last season.
Marca claimed the fee paid will be in the region of €45m, after a report initially broke the news saying that Madrid would be expecting a €50m move.
The 25-year-old made 51 first-team appearances for the Europa League winners since last summer’s loan deal, but there was no option to buy included in the deal.