The 24-year-old Old Trafford keeper has just a year to run on his current contract (via the Daily Mirror) and Man United have previously demanded £35m for the Spanish international (via the Daily Mail).
The Marca & the Metro (quoting Spanish sources) report that both sides want a deal completed this week.
Man United begin their Premier League campaign against Tottenham on Saturday and Louis van Gaal has already brought in Argentine international keeper Sergio Romero but has also been strongly linked with an interest in Ajax keeper Jasper Cillessen (via the Daily Mail).
David De Gea has been one of Man United’s most consistent performers over the past two seasons and has often been the club’s last line of defence, protecting a distinctly shaky back-line.
Sir Alex Ferguson brought in David De Gea in an £18.9m deal back in 2011 (via BBC Sport) and the Madrid born keeper did initially struggle but has matured into one of the best goalies in the game.
Louis van Gaal has already invested heavily in the summer transfer window having completed deals to sign Memphis Depay, Matteo Darmian, Bastian Schweinsteiger, Morgan Schneiderlin and Sergio Romero.
Man United are also close to agreeing the sale of Angel di Maria to PSG with the Argentine set for a medical with the Parc des Princes side (via BBC Sport).