The 32-year-old is due to see his contract expire at the end of this month, but the Red Devils have confirmed that talks are "open" with the Spaniard over a new deal.
"In terms of the goalkeeping department, discussions remain open with David de Gea over a potential new contract, while Tom Heaton and Nathan Bishop have had extensions triggered on their deals," a statement on United's official website read.
United had the option to trigger a one-year extension to De Gea's contract, but they decided against activating that clause, which expired in May, as they are keen to negotiate a longer-term contract which would see the Spaniard's wages significantly lowered.
De Gea has expressed his desire to remain at Old Trafford, and manager Erik ten Hag stated at the end of last month that he is confident the shot-stopper will pen fresh terms and commit his future to the club.
However, according to The Mirror, De Gea's contractual situation is looking "bleak" as the goalkeeper is so far unwilling to lower his £375,000-a-week salary.
Meanwhile, ESPN claims that the Red Devils' stance over De Gea's future has changed over the last month and they are "not as optimistic about reaching a compromise".
Man United have just two weeks to resolve De Gea's future or he will leave Old Trafford as a free agent on June 30.
De Gea has made a total of 545 appearances for the Red Devils across all competitions since joining the club from Atletico Madrid in 2011, the fifth-most in the club's history, and has kept 190 clean sheets.
Despite winning the 2022-23 Premier League Golden Glove award after keeping 17 clean sheets in 38 league games, United's No.1 was criticised on several occasions after making a series of costly errors leading to goals.
De Gea has made a total of 17 mistakes leading to goals during his Premier League career, while only Everton goalkeeper Jordan Pickford (13) has made more errors in the top flight than the Spaniard (11) since 2018.
Should De Gea fail to sign a new contract, Man United will step up their pursuit of a new number one this summer, with the likes of Inter Milan's Andre Onana, Brentford's David Raya and Porto's Diogo Costa all believed to be on the Red Devils' radar.
Onana has also been strongly linked with Chelsea this summer, but the Cameroon international recently reaffirmed his commitment to the Champions League finalists.
Raya, meanwhile, is believed to be close to agreeing personal terms with Tottenham Hotspur, although Ange Postecoglou's side are currently unwilling to meet the Bees' £40m asking price of the 27-year-old, which United are also not prepared to pay.
However, it is thought that United are prepared to activate Costa's €75m (£64.8m) release clause in his Porto contract, but the Portugal international has played down a possible summer move to Old Trafford.
Meanwhile, Man United have also revealed that Mason Greenwood has been included on the club's retained list despite remaining suspended after being charged with attempted rape, controlling and coercive behaviour and assault occasioning actual bodily harm.
The 22-year-old is currently awaiting the outcome of an internal investigation into whether he was in breach of his contract and brought the club into disrepute.