The 23-year-old star has shot to the top of Mauricio Pochettino’s wanted list in the last few days.
Kovacic, who has 36 caps for his country, played 27 games in La Liga for Real last season.
But he was in and out of the side and Spurs believe the lure of regular first-team football could tempt him into a move.
Real boss Zinedine Zidane wants Kovacic to stay, and will insist on a buy-back clause if any deal is struck.
The former Inter Milan star is under contract until 2021 and has also been linked with a return to Italy.
But Spurs are ready to move as doubts remain about the future of Eric Dier, who is wanted by Manchester United.
Kovacic would be an ideal replacement for Dier should he quit, but Pochettino also likes the player’s versatility.
He has played as both a defensive and attacking midfielder during his career and can even play in a wider role.
Spurs have so far had a quiet transfer window and missed out on Matthias Ginter, who quit Borussia Dortmund for Borussia Moenchengladbach instead.
But it is understood they are not interested in Monaco star Thomas Lemar, who looks set to join Arsenal.
Tottenham’s transfer plans have been thrown up in the air by Manchester City’s pursuit of Kyle Walker.
But they are determined to make a splash to reinforce a squad which finished second in the title race last season.
Spurs suffered with injuries at times and have made Moussa Sissoko and Kevin Wimmer available for transfer this summer.