While the brashly confident Mourinho - the self-proclaimed "special one" - foments confrontation, the studious and scrupulously polite Pellegrini, known as "The Engineer", quietly gets on with the job without putting noses out of joint.
If outspoken Portuguese Mourinho thrives in the spotlight and invites controversy, cultured Chilean Pellegrini prefers as far as possible to remain in the background, allowing his players to bask in the glow of success but assuming responsibility in times of failure.
Known for his ability to foster harmony in the dressing room and inspire a fierce loyalty among his players in the manner of a Vicente del Bosque or a Diego Simeone, Pellegrini has won wide praise for his man-management skills.
"I think his work speaks for itself," Brazilian forward Julio Baptista, who worked under Pellegrini at Malaga the past three seasons, told Reuters in February.
"Wherever he has worked he has managed to create very strong groups, not just of football players but of people as well, which is also very important," he added.
"He managed to create a strong group here at Malaga out of practically nothing."
Many of Mourinho's current and former charges also speak highly of the Portuguese's ability to inspire and motivate but the divisions that fractured the Real squad last season are one reason why he is on his way.
Pellegrini appears to have a knack of getting the best out of difficult characters, like unpredictable Argentine Juan Roman Riquelme at Villarreal, while also being able to nurture emerging talent such as Isco at Malaga.
He also has a steely side, at one point dropping the once-untouchable Riquelme after his form tailed off.
A former centre back with his sole club as a player, Universidad de Chile, Pellegrini later spent 16 years coaching teams in Chile, Ecuador and Argentina.
After moving to Spain, the gravelly-voiced 59-year-old won widespread admiration for his work at Villarreal, the modest La Liga club with a limited budget he joined in 2004.
He led them to third place in his debut season, when they qualified for the Champions League for the first time, and they also reached the quarter-finals of the UEFA Cup that year.
The following season was a breakthrough for both Pellegrini and Villarreal as they made it to the last four of the Champions League before losing 1-0 on aggregate to Arsenal.
Pellegrini's success landed him the ultimate prize in Spanish football and he was appointed Real coach in 2009 on the recommendation of then-club director general Jorge Valdano, a former Argentina and Real player as well as a former Real coach.
Notoriously impatient Real president Florentino Perez sacked Pellegrini after just one trophyless season, even after they amassed a club-record points haul of 96 in finishing second in La Liga.
Hired by Qatar-owned Malaga in November 2010, Pellegrini has proved his success at Villarreal was no fluke.
Following some hefty spending on players, the Andalusians qualified for the Champions League for the first time and came within a whisker of making the last four last term before falling to Borussia Dortmund.
Pellegrini's teams are admired for playing some highly entertaining football while at the same time boasting a mean defence, with powerful centre backs strong in the air and pacy full backs under orders to surge forward up the wings.
"Besides the fact that we have a group of extremely talented players, we have a very capable coach," Malaga's U.S. centre back Oguchi Onyewu told Reuters in November last year.
"He's pretty much a technician and he knows what he's doing," added the former AC Milan player.
"The team is comfortable and confident with his style of play and our cohesiveness is really the reason that we've been able to get some good results."
Ultimately, Pellegrini decided to leave Malaga because of uncertainty about the commitment of Qatari owner Sheikh Abdullah al Thani.
He had become frustrated with the Sheikh's apparent lack of ambition, underlined in January when Malaga sold Spain full back Nacho Monreal to Arsenal.
"He is the best coach I have had, the one I learned the most from," Monreal told Spanish sports daily Marca last month. "He knows everything."
Building a successful team out of a hugely talented squad packed with outsized egos is a challenge Pellegrini will relish and he will be sorely missed at Malaga.
Fans gave him a rapturous reception after the team's final La Liga match of the campaign at their Rosaleda stadium last month, a 3-1 win against Deportivo La Coruna.
He was awarded a prestigious gold medal by the city and will have a square named after him near the Rosaleda.
"Thanks to the work of Pellegrini and his team, Malaga was able to surprise the heavyweights of Europe," club director Francisco Martin Aguilar said last month. "And it's all thanks to this magnificent orchestral conductor."