The Paris Saint-Germain coach is said to have reached a verbal agreement with Real to take charge at the Bernabeu this summer with Mourinho being tipped for a return to the Premier League in England.
Paris newspaper Le Parisien claims the former Juventus, Milan and Chelsea manager Ancelotti is keen on the move to manage Real with his future at PSG up in the air.
Ancelotti has yet to sign a new deal with PSG despite leading them to the top of Ligue 1 and the last 16 of the Champions League.
Ancelotti is one of only 17 coaches to win the European Cup on two occasions - twice with Milan in 2003 and 2007.
His Milan side claimed the Serie A title in 2004 and the Club World Cup three years later while he led Chelsea to Premier League and FA Cup glory in 2010.