The former Chelsea boss is now expected to replace Julen Lopetegui this week after the embarrassing 5-1 defeat to their big rivals.
The writing was already on the wall for Lopetegui who has struggled since replacing Zinedine Zidane this summer, but the Clasico defeat was the final straw.
And the Guardian report that Conte was in talks with Perez on Sunday and is keen to take the job.
Conte still needs to finalise his compensation package with Chelsea after being sacked in the summer but that deal is likely to accelerate this week.
Lopetegui is expected to be sacked today with Santiago Solari in line to take over as caretaker until Conte can be appointed.
Real Madrid face lowly UD Melilla in the Copa del Rey on Wednesday night and then Valladolid in La Liga on Saturday.
Lopetegui was sacked by Spain on the eve of the World Cup after agreeing to take the Real Madrid while preparing for Russia.
And his time at Madrid is already over after they dropped to ninth after just four wins from 10 games.
Madrid are already seven points behind league leaders Barcelona, who themselves have not been completely on fire with three draws and a defeat in their opening 10 fixtures.
Madrid have failed to recover from the departures of Zidane and Cristiano Ronaldo, who joined Juventus in a £105m deal.
Mariano was the only forward brought in to help Lopetegui with Karim Benzema and Gareth Bale trying to make up the difference.
But Benzema has four goals while Bale has three so far this season while Ronaldo has seven for Juve.
Former boss Zidane meanwhile is the favourite to replace Jose Mourinho at Manchester United should he be sacked.
Former Real Madrid winger Steve McManaman admits he was stunned the club didn’t replace Ronaldo and thinks that has caused the problems this season.
“Everybody was surprised, weren’t they?” McManaman told Reuters.
“They took a long, long time to bring in new signings. They didn’t bring anyone to replace Cristiano.
“It was a big shock for everybody. Whether they could buy somebody of his quality I don’t know.
“The fact that they didn’t bring anybody in and they did slow business in the centre forward area, it surprised everybody.”