But Perez - newly elected as Real's president - insists transfer business has been on hold throughout the election process and they have not made offers for any player, including AC Milan goalkeeper Gianluigi Donnarumma.
Many were sure the Spanish giants wanted a move particularly for De Gea, not only because he is an outstanding shot stopper, but also because he is Spaniard, who could be an ideal long-term player and probably replacement for Casillas.
The Real president also spoke about Italy's 18-year-old goalkeeper Donnarumma, who is nearly certain to leave his Serie A club in the coming months.
'We haven't spoken to De Gea, ' Perez told Onda Cero. "Two years ago, we had an issue and then the appropriate circumstances didn't arise", Perez added.
'The players are very fond of Keylor and also the coach. "I love Keylor Navas and the truth is that, back in 2015, I didn't know him as I do now".
So can we finally put this rumour to bed? "Zidane's liked him a lot for a while, but it's not true we were close to signing him [a few years ago] because he still hadn't turned 16". 'We are following him, of course'.
The now 18-year-old has established himself as Milan's first-choice keeper with a number of excellent displays over the last two years, but agent Mino Raiola says that the club have not been treating their young star "like an asset".
The Spain worldwide nearly joined Madrid in 2015 until the blockbuster transfer collapsed and he continues to be linked with a switch to the Santiago Bernabeu, but Perez insisted the capital club are happy with Navas.