According to Simon Stone of BBC Sport, the Red Devils believe they are in a stronger position to resist Real now than they were two years ago, when only an administrative error prevented De Gea leaving for the Santiago Bernabeu.
The report added that senior Old Trafford sources said the Spain international has not requested a move "or expressed any discontent with his present situation."
De Gea's move to Real for £29 million was all but confirmed in the summer of 2015, but the deal collapsed at the last minute as the necessary paperwork was not submitted in time, per Stone.
The Spaniard subsequently penned a new four-year deal at Old Trafford to 2019—with an option to extend another year—and he has remained one of United's best and most important players.
The 26-year-old is arguably the best goalkeeper in the world, so it is no surprise United are eager to keep hold of him.
United's UEFA Europa League triumph, which ensured them a spot in next season's UEFA Champions League, could have played a big role in persuading De Gea to stay, per Sky Sports News HQ's Kaveh Solhekol:
Kaveh Solhekol @SkyKaveh
Manchester United confident David De Gea is staying at Old Trafford this summer. He's happy at United & looking forward to Champions League2017-5-30 22:14:29
Had the Red Devils not beaten Ajax in last week's Europa League final, and so failed to make the Champions League, it seems likely De Gea would have been left considering his future, especially given his potential prospects at Real.
According to the Manchester Evening News' James Robson, United have prepared contingency plans in case they cannot keep hold of their No. 1:
James Robson @JamesRobsonMEN
United believe they can keep hold of De Gea - but have had a contingency plan drawn up for some time with potential replacements2017-5-28 08:46:12
But it is clear United are prepared to make life difficult for Real should the Spanish giants pursue De Gea, and the stopper himself is reportedly happy to stay on at Old Trafford.