The club's incoming coach, Xabi Alonso, is said to be a big fan of the Argentina international and has given his approval for the move.
Xabi Alonso is set to take over as Real Madrid's coach at the end of the current season, after leaving Bayer Leverkusen.
Romero has been linked with other clubs, including Atletico Madrid, and has expressed his desire to play in La Liga.
Real Madrid previously tried to sign Romero last summer, but were unsuccessful. They are now planning to make another attempt.
Romero's contract with Tottenham runs until 2027.