The Portuguese worldwide is believed to be angered by the new offer from Madrid.
Real are back in action this weekend, with the La Liga champions travelling to Atletico Madrid.
The highly rated winger was, nonetheless, offered a new deal by the club, but he reportedly wants to leave at the end of the season, as he does not feel valued at the Bernabeu.
Prior to Ronaldo's announcement, the Spanish media had reported that Rodriguez had been admitted to Hospital Quiron Universal in Madrid to give birth.
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Spanish TV show El Chiringuito de Jugones claims that the 32-year-old has already informed Madrid president Florentino Perez of his decision.
Ronaldo is believed to want Chelsea's Eden Hazard to come in instead, allowing him to move in-field with the Belgian presumably deployed on the left.
"It's incredible to win an individual award in a team sport and also one where everybody wants to be able to say: 'I'm the best.'", Kaka said. "We are all very happy!" the football superstar wrote in Portuguese on Instagram, alongside a picture of himself together with Rodriguez and his first child, 7-year-old Cristiano Ronaldo Jr., in a hospital bed.
The Portugal worldwide has undergone a dip in form in recent months, scoring just one La Liga goal so far this season.