The Spain international is out of contract next summer, meaning De Ge could leave Old Trafford on a free transfer.
De Gea, 28, has been linked with a move away from United, with Juventus leading the race to sign Jose Mourinho’s No 1.
Manchester United have an option to extend De Gea’s current deal by 12 months.
However, according to The Metro, the Red Devils are reluctant to trigger the clause because they feel it will result in De Gea quitting the club.
They feel agreeing to the one-year extension is accepting defeat that the former Atletico Madrid star will walk away from United.
The Red Devils were confident of agreeing a new deal with De Gea, however, the Spaniard’s Old Trafford future is currently up in the air.
The Metro say United’s poor start to the season and Mourinho’s uncertain future at Old Trafford has played a keep part in De Gea’s thinking.
The Red Devils have set a deadline of late December before deciding whether to trigger De Gea’s 12-month clause.
United will be forced to agree to the extension, otherwise De Gea can sign a pre-contract agreement with foreign clubs in January.
The Metro add that United are willing to make De Gea the second-best paid player at Old Trafford, behind Alexis Sanchez.
Mourinho has been vocal on De Gea’s United future, with the Special One revealing he isn’t confident of keeping the Spaniard at Old Trafford.
“I am not confident,” he said.
“I cannot find the word in English, but let’s see what happens.
“Only the club, David and his people can answer that.
“My comment is only that everyone knows how good he is, how important he is for Manchester United and, if the club want to be better than it is and not worse, obviously it would be very important to keep David.”
United are back in action on Sunday afternoon, with Mourinho’s men facing City in the Manchester derby.