De Gea and United are back in action on Saturday afternoon when the Red Devils host Liverpool in the Premier League's early kick-off.
The 27-year-old still has 18 months remaining on his current contract and while that still remains the case, Real will continue to test the water to see if they can convince the 27-year-old to return to Madrid.
United are keen to hand the Spaniard a new deal, with a salary matching his status as the best goalkeeper in the world.
The Spanish keeper has been United's standout performer this season and is regarded as one of the best in the business.
De Gea has been a target for Madrid for a number of seasons now. His influence can be quite characterised by his incredible save that he pulled against Crystal Palace on Monday - an nearly inevitable goal, which was one of the main reasons United won the game 3-2.
The Portuguese boss was adamant that the club will not sell De Gea, though, amid Real interest reports.
Real Madrid's defence has been woeful this season and Keylor Navas has been found out largely because of his inability to pull off consistently good performances.
Sport, De Gea and his representatives are looking for the club to match the talk from Jose Mourinho over his fine form.
Athletic Bilbao's Kepa Arrizabalaga, 23, has been touted as a potential replacement, but he signed a new agreement at the San Memes, while Thibaut Courtois' situation at Chelsea has also been monitored.