Real promoted Solari to the hot seat following Julen Lopetegui’s ill-fated tenure at the club.
However, the club hero has struggled and Los Blancos slumped to a shock 2-0 defeat against Real Sociedad on Sunday.
They remain 10 points behind pace setters Barcelona.
Mourinho, meanwhile, was dismissed by Manchester United following a dreadful run of results and disquiet behind the scenes.
Benfica made a move to secure his services, but the Portuguese remains unemployed – for now.
He spent three years at the Bernabeu from 2010 – 2013 and is still held in high regard.
Journalist Neil Custis claims Florentino Perez is already making overtures.
He told talkSPORT: “He’s been offered about four jobs, one of which was Benfica, but also Real Madrid have made contact with his camp about how he feels about a return to the Bernabeu.
“Obviously Real are struggling, 5th I think they are – 10 points behind Barca and are on their second manager of the year.
“The president there, [Florentino] Perez, has remained friends with Mourinho even after he left the last time.
“He didn’t want him to go, player power behind the scenes made him go.
“But he was a very successful manager at Real Madrid.
“You know, to overcome that Pep Guardiola team – to win La Liga – was something special.”
Spanish news outlet COPE say Solari will remain in post until the end of the campaign.
However, there are clues that the writing could be on the all for the 42-year-old.
It’s suggested he will not be allowed to make any more signings.
Brahim Diaz has been drafted in from Manchester City for around £21m.
The midfielder had made just 15 appearances for the Citizens.
City boss Guardiola says he wasn’t bothered about losing the 19-year-old.
“Hopefully other players won’t decide to move on,” said Guardiola.
“I don’t want people who don’t want to stay here, to be with us and try to achieve what we want to achieve.
“If players want to stay, the door is always open for them.”
Daily Express :: Sport Feed