The offer, worth €65m, was made in the past 24 hours, according to sources close to Real Madrid.
Rodrygo, a key player for Real Madrid, was initially hesitant to join Tottenham due to the club's reputation and recent successes, including winning the Europa League.
However, the player's stance may have changed in recent weeks, possibly due to a lack of more attractive offers from other top European clubs.
Tottenham have already asked Real Madrid for permission to negotiate directly with Rodrygo, a move that could potentially lead to a transfer.