The Spanish goalkeeper's long-term future at Old Trafford is in question with the 31-year-old in the final year of his contract.
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United have the option of extending De Gea's stay by an extra 12 months, with the ex-Atletico Madrid man keen to remain at the club.
But according to The Athletic, he is not a natural fit to Erik ten Hag's style of play of playing out from the back, leading United to look for other options.
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It has been said that United would be prepared to hand fresh terms to De Gea on a lower wage, while bringing in a new goalkeeper to challenge him between the sticks.
United's chief goalkeeping scout Tony Coton has been travelling around Europe over the last few months, with Costa and Simon catching his eye.
The Premier League outfit are keen to recruit a shot-stopper who can play for United for the next decade.
Costa has been the standout goalkeeper in the Champions League this season, where he created history on Wednesday.
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The 23-year-old saved his third consecutive penalty in the group stage by keeping out Bayer Leverkusen striker's Patrik Schick spot-kick, before saving Kerem Demirbay's effort in the return match a week later.
And on Wednesday night he denied Noa Lang from 12 yards out TWICE, as he saved his first penalty before the winger's retaken spot-kick shortly after.
But more importantly, he is comfortable with the ball at his feet and is adept at starting attacks out from the back.
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Simon, meanwhile, is also on United's radar with the 25-year-old set to start ahead of De Gea for Spain at the World Cup in Qatar.
Should either Costa or Simon be brought in, it would likely see Dean Henderson depart as he would only return to United to be Ten Hag's no.1.
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The England international is currently on loan at Nottingham Forest where he has appeared in all 12 of their Premier League games.
Speaking earlier this month, Ten Hag refused to comment on De Gea's future at United.
He said: "We didn't make a decision on him but what I can say is I'm really happy with David De Gea as a 'keeper.
"The first two games of the season were not his best performances but after it, you see he brings us a lot for the defence, the defensive department, he gives stability and, as a person also, I like him.
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"I can cooperate really well with him but first we go to the winter and then we will talk about how we deal with situations."