And the 47 year-old has wasted no time pinpointing the positions in which he will be looking to bolster his squad over the summer.
While praising the ability of Marcelo, who currently plays at left-back for the Bernabeu-based club, speculation suggesting that Mourinho and Cole may be reunited next season is rife and the former Chelsea manager's comments will do nothing to calm the whispers.
"I would like to have an alternative left-back to Marcelo," Mourinho said. "Marcelo has his characteristics but Real needs a balanced squad, different players for each position. I would like a left-back that has different characteristics to Marcelo.
"Real has a right-back that I like very much in Sergio Ramosm but in my way of thinking, I believe he can be better as a central defender so we might need a right-back.
"Real must reach the final of each competition, regardless of whether we win or not, and for that reason, you must have different players for each position.
"Right now, I have a young team with quality players. Real always looks for excellence and they find quality players."
Madrid's midfield has been considered by the club's fans throughout the season as their biggest weakness and is another area high on Mourinho's list of matters to address.
"The midfield line is very important," Mourinho said. "It's difficult to find a complete player in central midfield. You have less players box to box. You have more number 10s and defensive midfielders.
"But it's very difficult to find a player in midfield right now that is complete."
As to who the best midfielder in the world is, Mourinho said: "It's difficult to choose who the best is because they play in different football cultures.
"You can't say that (Chelsea's Frank) Lampard will score 30 goals in Serie A or in the Primera Division but in England he averages 28 goals per season, which is incredible.
"You cannot say either that (Barcelona pair) Xavi or Andres Iniesta will perform with the same quality if they were to go to the Premier League or Serie A."
One midfielder who looks to be on his way out of the Bernabeu is Guti.
The 33-year-old announced a month ago his wish to leave the club after 15 seasons, only to change his mind three weeks later and say he wants to see out the remaining year on his contract.
"I like Guti because he has talent," said Mourinho. "But if a player said a month ago that he had played his last game for Real, he cannot come a month later and say that he wants to stay.
"I like players that are committed to the club, the project, his team-mates and the coach and I don't like this doubt."
Mourinho's direct style does not make him universally popular, but he does not believe it necessarily makes him a difficult coach to work with.
"It can be very easy or very difficult to work with me," he said. "It's true that I see things black or white and those that are not with me are my enemies only in football terms as I respect everyone in life.
"If someone works hard and is aware that the club is more important than all of us, then it's very easy to work with me.
"If a person is egocentric and selfish, has no passion for football and doesn't like to work, it's difficult.
"Practically all my players have adapted well to me and, fortunately, I've been lucky in that respect because that has allowed us to win titles."
Mourinho has been brought to the Bernabeu by president Florentino Perez in a bid to bring the good times back to the Spanish capital, with Barcelona having won La Liga for the past two seasons.
Arriving on the back of a hugely successful stint with Inter, the former Porto and Chelsea manager insists he is not worried by the expectations on his shoulders.
"My pressure is Mourinho on Mourinho," he said. "That is what I need - not the pressure of the press or the fans. You have to put pressure on yourself, to be the best and want more and more from yourself.
"If the day comes when you accept that you don't care about winning or losing any more, and you don't want to be the best, then it's time to go home and not work."