The Portuguese manager has signed a three-year contract and took charge of training on Monday as the club began pre-season preparations in Madrid.
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Mourinho previously led Real from 2010 to 2013, delivering success while dividing opinion with his intense and controversial style.
He now faces the challenge of reviving a struggling side that has failed to win a major trophy for two years despite having star striker Kylian Mbappe.
Madrid endured a turbulent season, changing coaches twice with Xabi Alonso and Alvaro Arbeloa both taking charge.
President Florentino Perez’s decision to call and win early elections created further uncertainty, but his victory cleared the path for Mourinho’s high-profile comeback.