According to Paul Hirst of the Times, the Spain international is ready to sign a new contract at Old Trafford, with his current deal poised to expire in 2019; United do have an option to extend his stay by another year.
"Real Madrid are desperate to sign a goalkeeper this summer, and although they have made it clear that they would be willing to sign Thibaut Courtois, of Chelsea, they also retain an interest in the United player, who has been a long-time target for the European Cup holders," noted Hirst.
However, according the report, Los Blancos are now "likely to fail" with any offer for the former Atletico Madrid man, with talks triggered between United and De Gea about a new contract.
As relayed by Ciaran Kelly of the Manchester Evening News, according to Hirst, the discussions have "progressed well," with United willing to pay the 27-year-old more than £220,000-a-week as part of his fresh terms.
Per Simon Stone of BBC Sport, manager Jose Mourinho made it clear in January he wants to keep De Gea around:
That's no surprise whatsoever, as the United No. 1 continues to establish himself as a world-class talent.
During his time in Manchester, the stopper has made rapid progress, and he is now the Red Devils' most important player. De Gea is an incredible athlete, a tremendous shot-stopper and someone who makes major contributions for United in clutch moments.
That's been the case in recent years, and earlier in the week, he proved to be decisive again. In the 0-0 draw against Sevilla in the Champions League round-of-16 first leg on Wednesday, De Gea somehow kept out Luis Muriel's close-range header to preserve parity in the game.
Former Sunderland goalkeeper David Preece was effusive in his praise for the Spaniard following the crucial stop:
De Gea has been perennially linked with a move to Real Madrid, and Los Blancos did come close to signing him in the summer of 2015 before a move collapsed in the final embers of the summer transfer window.
Speculation has persisted since, although it's rarely been as intense as it was prior to that failed move. Keylor Navas has developed into a dependable presence for the capital club, doing a fine job as Madrid won the last two Champions League tournaments in succession.
Even so, De Gea would represent an upgrade on the Costa Rican, and Real are renowned for chasing the world's best players.
United legend Rio Ferdinand said De Gea is exactly that in midweek:
The temptation for a Madrid native to move to the Santiago Bernabeu would be understandable, and as a result, United fans would feel much more settled if a new deal was to be struck.
De Gea continues to be crucial for the Red Devils and is an irreplaceable presence at the base of the team. Should he commit his long-term future to the club, the onus will be on others at United to build a side ahead of him capable of challenging for major honours consistently.