The flamboyant Portuguese helped Real end six years of Champions League failure on Wednesday when they beat Olympique Lyon 3-0 for a 4-1 aggregate success to claim a berth in the last eight for the first time since 2004.
That was the year Mourinho won the Champions League with Porto before moving to Chelsea and then to Inter Milan, whom he led to a treble of Italian league and cup and Champions League last season.
"I prefer to avoid Chelsea and Inter because emotionally it's difficult playing against your people and your friends," Mourinho told a news conference.
"If we have to play them then we will play them as we are all professionals but I'll have to forget they are my people.
"Emotionally I will have to better prepare myself to play Inter or Chelsea."
Real could also be drawn against great Spanish rivals and 2009 winners Barcelona.
They are five points behind Barca in La Liga with 10 games left and are due to play them in the King's Cup final on April 20.
Real Director General Jorge Valdano said it would be better for Spanish football if the pair met later in the competition, adding that he would not like to draw Schalke 04, as the return of former Real striker Raul would be too much of a distraction.
"We wouldn't like it if Raul visited us," Valdano said in a television interview.
"There would be too much emotion in the atmosphere and it wouldn't be advisable."
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