The capital club's honorary president says the Portuguese coach was excessively defensive against Barca, and warned him that Madrid have axed more successful trainers in the past
Mourinho was jeered by his own fans in Sunday's 4-1 win over Athletic Bilbao, having lost 2-1 at the Santiago Bernabeu to Barca four days earlier in the sides' Copa del Rey quarter-final first-leg clash.
The Portuguese coach was heavily criticised for his ultra-defensive gameplan in that match, and the club received more bad press for Pepe's cynical stamp on Lionel Messi, which went unseen by the referee.
Reports of a training ground bust-up between Mourinho and defender Sergio Ramos followed and television show Punto Pelota claimed on Sunday that the Portuguese would leave Madrid in June.
And Di Stefano has now warned the coach that he may be treading on thin ice at the capital club.
"The spotlight is on Mourinho," he wrote in his column in Marca.
"Madridismo didn't like his plan at the Bernabeu ... for the first time the fans launched their reproach against Mourinho. Deep down [it was] the memory of the fiasco against Barca and especially the rejection of a type of behaviour many consider degrading to Madrid."
Di Stefano, who has been critical of Mourinho's tactics in the past, also warned the Madrid boss that more successful coaches have been dismissed in the past at the Bernabeu.
"Jupp Heynckes was sacked after ending a 32-year European Cup drought. Vicente del Bosque wasn't kept on after the conquest of two Champions Leagues titles and two leagues," he remembered.
"The same thing happened with Fabio Capello, and the sacking of [Radomir] Antic came when the team were top of the league.
"The history of Madrid has been written that way dozens of times."
And the Argentine legend believes now is the time to deliver for the Portuguese boss.
"Mourinho must get it right," he wrote.
"Up until now he hasn't found the formula [against Barca] and he confessed that after the game at the Bernabeu.
"He paid the consequences for [an] excessive defensive [gameplan].
"Perhaps at Camp Nou [the team] will rise to the greatness the history of Madrid demands."
Mourinho led Madrid to victory in last season's Copa del Rey, while his side currently lead La Liga by five points from Barcelona.
Starting in November and continuing through to the middle of February, Goal.com is publishing exclusive extracts from Graham Hunter's highly-anticipated new book, 'Barca: The Making of the Greatest Team in the World'. Friday's extract looks at how Barca suffered rare disappointment at the hands of Real Madrid in a gripping finale to last season's Copa del Rey. Don't miss it!