If he can do that, impatient Real fans and management will be prepared to overlook the methods by which it may be achieved as success in the Champions League has become their Holy Grail.
Real won Europe's top club competition three times in six years between 1998 and 2002, made the semi-finals in 2003 and the quarters the year after, but have failed to progress beyond the last 16 since.
NEWS:Real unveil Mourinho
Last season's surprise exit to Olympique Lyon, their sixth successive defeat in the first knockout round, sealed Manuel Pellegrini's fate, while failure to reach the final at their own Santiago Bernabeu stadium was particularly galling.
When Real president Florentino Perez authorised the expenditure of 250 million euros on new players last year, that had surely been one of his objectives.
Speaking after his sacking last week, Pellegrini said he did not believe winning La Liga would have made any difference to his future in the post, and that was after his team scored 102 goals and smashed the Spanish points record.
Barca went one better to finish three points ahead on the final day of the season, with their home and away victories over Real in the clasicos' the difference between the two giants.
Mourinho, of course, out-manoeuvred Pep Guardiola's side to knock them out of the Champions League at the semi-final stage, and he will be expected to do something similar next season when the sides meet in the league.
Barca have not rested on their laurels and have snapped up Spain striker David Villa to strengthen their side. Real have yet to move in the transfer market but this could be the first point of friction between Mourinho and his new bosses.
Pellegrini, as other Real coaches have before him, complained of being isolated from the planning and decision-making processes at the club, saying they undermined the position of the coach.
Mourinho is likely to want more control over who is signed and will fiercely resist any kind of interference in team affairs from Perez's right-hand man Jorge Valdano and sports director Miguel Pardeza.
The Portuguese is the 10th coach hired in eight years since Vicente del Bosque led them to their last European triumph in 2002, and he has to please the suits upstairs as well as a demanding fan base, many of whom are suspicious of Mourinho's reputation for being a defensive coach.
Real fans expect their sides to entertain and want to see the likes of Cristiano Ronaldo and Kaka cut loose. The atmosphere at the Bernabeu resembles that of a bull-ring where a matador is judged by how well he kills the animal, his style, his daring and his flourishes.
Keeping a Galactico like Kaka on the bench, for example, will test the patience of Real's marketing men and the fans, but they will perhaps forgive him if he leads them to next year's Champions League final.
Three European Cup triumphs with thre
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