Real Madrid came within minutes of signing De Gea from Manchester United in 2015, but a faulty fax machine saw the deal fall through.
Los Blancos’ interest in the 27-year-old appeared to have cooled recently, with Thibaut Courtois and Kepa Arrizabalaga wanted at the Bernabeu instead.
However, Spanish website Don Balon say Real are still determined to sign De Gea from United despite the two clubs’ strained relationship.
Yet Manchester United are supposedly only willing to entertain an offer for the shotstopper if Real include Asensio in any negotiations.
The youngster could leave the club in pursuit of regular first-team football, despite having the backing of club president Florentino Perez.
Asensio starred last term as the club waltzed to the La Liga and Champions League titles.
But the 21-year-old has found life tougher this term, despite putting Barcelona to the sword in the Spanish Super Cup at the start of the season.
Cristiano Ronaldo and Gareth Bale are currently ahead of him in the pecking order and while the two players remain, Asensio will always be forced to play second fiddle.
Yet Don Balon say that, despite his struggles, he is admired in the Premier League by Red Devils boss Jose Mourinho.
Any transfer, however, could depend on who is in charge of Real come the end of the season.
Zinedine Zidane has steered the club to consecutive Champions League titles since taking the reins from Rafa Benitez in January 2016.
But that may not be enough to keep him in the job.
Germany coach Joachim Lowe and former Madrid boss Carlo Ancelotti have both been tipped to take over if Perez loses patience and wields the axe.
Real currently sit 16 points behind Barcelona in the La Liga table and a big summer is needed if they are to pip the Catalan giants to silverware next term.