And the Spanish giants are convinced they are favourites to land the Portuguese playmaker — who has long been a target.
However, Bernardo only signed a 12-month extension a year ago, stretching his current deal to 2026.
The deal also includes a £50million release clause.
And Bernabeu bosses are determined to outmuscle French outfit PSG.
It was the 29-year-old’s fluffed shootout spot-kick which helped Real wreck Manchester City’s dreams of a double Treble.
Yet they reckon his versatility in midfield would make him a key figure at the Bernabeu.
While the release figure is seen as a bargain for someone of his ability.
Bernardo is aware of the interest from Real and PSG.
And although there is no issue with City, he would prefer a move overseas for family reasons.
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Real aren’t the only Spanish giants interest in Bernardo though.
Barcelona are long-term admirers after attempting to land him last summer.
And striker Joao Felix, also Bernardo’s Portugal teammate, admits the City man has been asking about life in Catalonia.
Felix, on loan at Barca from Atletico Madrid, said this month: “If Bernardo Silva comes, I’m going to ask for a commission.”
“Everything I have said is true and if he comes they will be delighted with him.
“He is very careful with everything and he asked me about places to live, places to eat, about security.
“He asked me a little about everything and I said about tax issues because that’s what came to mind.
“I think so, I think so (Bernardo’s convinced by Barcelona). Well, but now I don’t think it depends on him.”