Modric has returned to training with Spurs this week, having been fined by the club for skipping sessions as he sets his sights on a move away from White Hart Lane this summer.
Manchester United and Paris St Germain have both been linked with the Croatia playmaker, with Tottenham believed to be holding out for a £40million fee.
But Real appear his likeliest destination, a move which would allow Brazil international Kaka to end a largely ineffective spell at the Bernabeu.
Big-spending PSG have also been mooted as the 30-year-old's next stop and he did not travel with Real for tonight's friendly away to Real Oviedo, which they won 5-1.
Karanka told AS: "We decided he would stay in Madrid. Kaka and the club are working to find a good solution for all parties."
Asked about Modric, he added: "Which club would not want a player like him? He is an outstanding footballer and had a great tournament (Euro 2012). It is clear that he would fit in Madrid."