The Portuguese has signed a four-year contract with the Spanish giants following his move from European champions Inter Milan.
"It's an honour to have one of the most prestigious coaches in the world," Real sporting director Jorge Valdano said.
"We are delighted to have him here with us. The best thing for our club right now is to have Mourinho with us."
The 47-year-old Mourinho, former coach of Porto and Chelsea, replaced Manuel Pellegrini who was sacked last week.