Atletico Madrid sprung a surprise by defeating Los Blancos 3-1 on Sunday night at the Metropolitano.
Former Real Madrid forward Alvaro Morata netted the opener inside four minutes, before Antoine Griezmann made it 2-0.
Toni Kroos reduced the deficit with a sublime strike before half-time, but Atletico regained their two-goal advantage when Morata headed home after the break.
Real Madrid were second-best throughout the contest against their city rivals, and Bellingham's frustration grew towards the end of the clash.
The England international produced a reckless challenge on Angel Correa, and was shown a yellow card by the referee.
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Simone was left livid post-match following the challenge, and claims it was worth of a red card.
"Jude Bellingham's challenge looked like a red card when I saw it live," he said.
"For me, Bellingham should have been sent off."
Even though the England star was playing against his new rivals, he showed his respect to them after the clash.
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The 20-year-old was pictured walking off the pitch at full-time, and avoided stepping on the Atletico Madrid crest in a show of respect.
Bellingham has had a bright start to life in the Spanish capital, most recently scoring a last-gasp winner against Union Berlin on his Champions League debut.
He has managed six goals in seven games for Carlo Ancelotti's side since joining from Borussia Dortmund over the summer.
The midfielder will be looking to add to his tally when Los Blancos take on newly-promoted Las Palmas at the Santiago Bernabeu on Wednesday.