Despite the Brazilian saying the "door is closed" on a move to the Santiago Bernabeu, the Liga champions have made a final bid to sign president Ramon Calderon's main transfer target.
Kaka's advisor Diogo Kotschko said: "Real Madrid is sure that this money can change Milan's mind, and I believe it is an opportunity that Kaka must consider very carefully."
Real have made no secret of their desire to sign the brilliant playmaker, whose form in helping Milan win the Champions League has seen him hailed as the best player in the world.
The Italian side have been angered by Real's tactics, as well as the mischievous Spanish press, who pictured Kaka holding a mocked-up newspaper front page hailing Real's Liga success.
The 90 million euro (£60m) fee would make the transfer a world record. Real hold the current record for the £46.5m they paid for Zinedine Zidane in 2001.
However, Milan president Silvio Berlusconi has claimed the club do not need to sell as they are in rude financial health, and could even afford to shell out 100 million euros for Barcelona star Ronaldinho.