De Gea endured a difficult first season with the Red Devils, making a string of high-profile mistakes early on before finding form as Sir Alex Ferguson's men just missed out on the title on goal difference.
And Bailey, who played for United between 1978 and 1987, feels that the Spaniard has the right men around him to be one of United's best-ever shot-stoppers.
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“David is still very young and will mature - which a goalkeeper needs to do quickly - but with Eric Steele and the manager working with him, he is potentially one of United’s great goalkeepers," he told the club's official website.
"There could come a time where we’re talking of him in the same breath as [Peter] Schmeichel or [Edwin] van der Sar.”
Bailey also expressed his belief the former Atletico Madrid keeper is finally beginning to adapt to the rigours of the Premier League following his patchy start.
He added: “De Gea is the one who has the jersey at the moment, and the important thing for him was that he showed a lot of improvement during the course of last season.
“In Spain strikers duck out of the way of the goalkeeper, but in England they almost look for an excuse to charge at you. It took him a little while to get used to that and understand it.
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"He recovered very well under a lot of pressure, stood up, came back and earned the right to be first choice.”
And, whilst Bailey insists that De Gea is rightly the first choice between the sticks, he also acknowledged that Ferguson currently has an embarrassment of riches at his disposal in goal.
“United are very well stocked; they’ve got three very good goalkeepers," the South African continued.
"Ben Amos has had a couple of games and I’ve been very impressed whenever I’ve seen him play. Anders Lindegaard looked really solid and reliable until he picked up an injury last season, and David de Gea has got a lot of talent."