Mourinho was sacked at Old Trafford in December and is currently available if Los Blancos ask him to return to the Spanish capital.
The move would represent a dramatic rise back to prominence for the coach after his disappointing departure from the Premier League.
Mourinho left Madrid in 2013 before taking the reigns at Stamford Bridge for a second stint.
The manager took Madrid to the La Liga title in 2010-11, winning the championship by nine points ahead of Barcelona.
Mourinho has recently spoken about his experiences working at Real, saying he would have no problem returning to collaborate with Madrid president Florentino Perez, per BeIN Sports' Mitch Freeley.
ESPN FC's Sid Lowe reported the pair have maintained a good relationship since Mourinho exited the Bernabeu.
The Portuguese told Deportes Cuatro (h/t Alex Richards of the Mirror) on Tuesday he has fond memories of managing the club: "I have a fantastic memory of working in Real Madrid. It is a unique experience and different to the rest of the clubs I have been at. We did fantastic things, we won the league in a unique way. We also had bad moments, I made mistakes, but after, the experience made me a better manager and person."
Real were dumped out of the UEFA Champions League on Tuesday after a stinging 4-1 home defeat to Ajax in the last 16.
The loss compounded the misery for Solari who has struggled to get the very best from his squad since taking over from Julen Lopetegui.
Madrid sit in third in La Liga, and the European failure could be the moment that sealed Solari's fate.