The exciting playmaker only made 10 starts in the league for Los Blancos last season, and has subsequently been linked to a number of clubs following the resignation of manager Zinedine Zidane, including Roma, Juventus and Inter Milan.
According to Spainish outlet AS, via the Daily Star, Manchester United were expected to sign the player for around £65m, where he would join new £52m signing Fred in midfield. However, with Mousa Dembele rumoured to be moving abroad, Tottenham Hotspur have attempted to hijack the deal and sign the unsettled Croatian.
Kovacic is unlikely to address these transfer rumours as he is away at the World Cup with the Croatian national team, who have managed to top Group D by winning all of their games, including an emphatic 3-0 win over Argentina.
The 24-year-old has been with Real Madrid since 2015, in which time the club has won three Champions League titles in consecutive seasons, as well as one La Liga crown.
The young Croatian is up against strong competition in the Real Madrid midfield, with Luka Modric and Toni Kroos often starting ahead of him.
Tottenham are yet to make a signing so far this summer, but have been linked to a number of players, including Aston Villa's promising young midfielder Jack Grealish and OGC Nice forward Alassane Plea.
The London-based club will be hoping to challenge for the title next season, as they have finished in the top three for the previous three seasons.