The ex-Manchester United goalkeeper praised the current No.1's "miracle saves", before warning that counter-attacking will suit the Spanish giants in the second leg
The former Atletico Madrid shot-stopper returned to his hometown and made a number of key saves in the first leg of the Champions League last-16 tie at the Santiago Bernabeu as United earned an important 1-1 draw.
The 22-year-old has received criticism for a lack of consistency since his €21 million move in 2011, but Bosnich says the "outstanding talent" on show on Wednesday night is De Gea’s real identity.
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"That's just a matter of practise and timing," he told talkSPORT. "What he's actually got and what you are working with is something very special.
"You saw that last night, they were miracle saves that he was making. They are the saves that win you a game or get you through a tie like this. He proved everyone wrong against Real Madrid. He is an outstanding young talent.
“He’s come to England at a young age and has had to adapt to the culture of high balls during games. It’s taken him a little bit of time to get used to it but he showed his true colours last night.”
Bosnich, who represented United 26 times during two spells, also praised Cristiano Ronaldo’s performance after the attacker netting Madrid's equaliser with a fantastic header after Danny Welbeck opened the scoring.
"When you watch it in slow motion, how high he got up, it was phenomenal," he continued. "It was a phenomenal leap and he's a phenomenal player.
"Lots of people don't like him because of the way he struts his stuff but no-one can deny what a fantastic player he is.
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"He showed his class last night and he's going to be they will have to watch very carefully at Old Trafford, coming back to where he made his name he'll be wanting to put on a really good show."
The 41-year-old then suggested that, despite United securing an away goal, Jose Mourinho's men may be better suited on the counter-attack which will make Sir Alex Ferguson's tactical approach tough for the return leg.
"If both sets of strikers have their boots on [the second leg] has the potential to be a high-scoring encounter," he added.
"The Manchester United faithful will be wanting them to go forward but if you watch Real Madrid, the matches they struggle in are when they have to make the play.
"They rely on counter-attacking, if you look at Jose Mourinho's teams at Inter Milan and even Chelsea they did like to play that so they can open up space in behind the defence and then counter.