The Portuguese manager was the first coach to give Rodriguez his chance in the Real Madrid senior side after promoting him from the reserves.
“I owe everything to Mourinho, he gave me a chance when I was at Castilla and he had enough faith in me to promote me to the A team from the reserves,” Rodriguez was quoted as saying by NTV Spor.
The 20-year-old midfielder was praised by Mourinho during his stint as manager of the Spanish giants.
The then Real Madrid boss stated: “José Rodríguez has the potential to become a Real Madrid first team player.”
The Real Madrid youth product spent last season on loan at Deportivo de La Coruna providing one assist and scoring two goals in 27 appearances.
The 20-year-old midfielder has represented Spain at U17, U19, U20 and U21 level.
Galatasaray won the league title for the 20th time last season and booked a place in the Champions League group stages for the fourth consecutive season.
Rodriguez is expected to face his former team Real Madrid in the 36th edition of the Santiago Bernabeu Trophy in Madrid on 18 August 2015.