Negotiations continue to rumble on between Real and Manchester United over a deal for De Gea.
The Spain goalkeeper's contract carries a €70 million buyout clause, but Perez is reluctant to go so high for a goalkeeper.
Instead, he hopes to drive down the cash component by including a player in his offer.
And after initial proposals were rejected by Mourinho, Perez is now offering one of Real's first-choice stars: Luka Modric, Toni Kroos or Karim Benzema.
For his part, Mourinho would be happy to take Kroos - though his priority is Raphael Varane.
Meanwhile, it's also emerged De Gea has demanded Keylor Navas be sold by Real before his arrival - something which has not gone down well inside the Real dressing room.