Manchester United and Inter Milan are also chasing the Croatia international who is one of seven players set to leave the Bernabeu next summer.
During his five-year spell in Madrid, Modric has lifted the Champions League three times and the La Liga title once.
Tottenham signed Modric during the summer of 2008 in a £16.5m deal from Dinamo Zagreb. During his four years at White Hart Lane, the Croat made 159 appearances for the club but endured a barren spell in terms of silverware.
He’s made up for that in Madrid. As well as La Liga and the Champions League, winner’s medals for two Copa Del Rey’s, the Supercopa de Espana three times, UEFA Super Cup three times and the FIFA Club World Cup twice.
Manchester United, however, believe the personal relationship between Modric and manager Jose Mourinho will sway the decision in their favour. The Portuguese was Real Madrid boss when Modric moved to Spain and set to miss out on Mesut Özil, is keen to bring another player he admires to Old Trafford.
The fly in the ointment is the interest of Inter Milan. Reports from Italy suggest Modric is keener on Serie A as he approaches the twilight of his career.