The Spanish giants failed to make home advantage count during their last-16 first-leg tussle with the Red Devils.
A series of fine saves from David de Gea kept Real at bay once Cristiano Ronaldo had scored against his old club to cancel out Danny Welbeck's first-half opener.
Trailing Barcelona by 16 points in La Liga, it is widely assumed Mourinho will find himself out of a job should the Spanish giants fail at Old Trafford in the second leg on March 5.
However, the former Chelsea coach insists he is not under pressure.
"I don't feel any pressure," he said.
"I work hard, am very serious and am happy with what I do.
"I have peace of mind and my head is high, but there is no pressure really.
"I have no idea (if his future depends on the second leg) and I am not worried about it."
However, Sir Alex Ferguson feels such an outcome would be very harsh.
"For Jose to be here as long as he has is unusual and a mark of respect for the work he has done," said the United boss.
"I don't know how many coaches they have had - 25? That's more than the years I have been here."
Given he is averaging over a goal a game in nearly four seasons with Real, it should come as no surprise that Ronaldo scored on Wednesday night, rising above Patrice Evra to meet Angel di Maria's cross with a powerful header.
But Ronaldo opted not to celebrate, preferring instead to acknowledge the Real fans before an almost apologetic gesture to the United bench.
"I was in Manchester for six years so tonight was very emotional for me," said Ronaldo.
"I scored a goal but I didn't celebrate out of respect.
"Maybe United are favourites now, maybe not. Anything can happen.
"Sometimes we are better away than at home so we will see.
"It will be amazing to play at Old Trafford again after four years. It will be another emotional night."
United now face Reading in the FA Cup on Monday, then have a trip to QPR and a home game with Norwich, giving them a relatively straightforward period until the deciding leg at Old Trafford.
In contrast, Real have two meetings with Barcelona.
"It was possible for them to win 3-1 because they had two big chances when Alonso cleared off the line and Diego made a save," said Mourinho.
"I won't criticise what they did. United did their job well and got the result they wanted.
"They came here not to concede space and goals and they didn't play to win.
"But they know Real Madrid can do it there."
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