The clubs' respective presidents, Massimo Moratti and Florentino Perez, met in Milan on Friday afternoon to thrash out a financial deal to facilitate the early release of Mourinho from his contract at the San Siro, which ran until 2012.
Mourinho will replace Manuel Pellegrini in the Madrid hotseat after the Chilean was sacked on Wednesday night.
He will be presented at the Bernabeu at 1pm local time on Monday.
A joint statement from the two clubs read:
"Massimo Moratti and Florentino Perez met this afternoon.
"In light of the excellent and constructive relations of friendship that exist between the two clubs, an agreement was reached on the contractual clause that binds Jose Mourinho to FC Internazionale.
"The time and the manner by which the agreement will be honoured has been defined.
"Massimo Moratti thanks Florentino Perez for his willingness to join him in Milan and for his courtesy, which confirms, once again, the strong link between the two prestigious clubs."
Inter president Moratti had earlier hinted compromise was not on the agenda.
"Mourinho has been spectacular, intelligent, brave and we acknowledge that," said Moratti. "This doesn't take away the fact that there is still an issue regarding the negotiations with Mourinho and with the club that wants him.
"We want to end Mourinho's negotiations well, but we have a right to defend a position that is based on honesty and legality with regards to an agreement that was done with someone (Mourinho) that did their duty and the club (Inter) that allowed him the chance to free himself of his contract.
"Our position is not one of force. There are conditions and I don't think our attitude has been harsh. We would have continued with Mourinho because there's no coach like him."