The Portuguese global is believed to be angered by the new offer from Madrid.
Ronaldo, whose current contract does not expire until 2021, is reportedly unhappy with Madrid's failed pursuit of Kylian Mbappe during the summer, with Los Blancos missing out to Paris Saint-Germain in the race for the highly-rated France global.
Following last season's Champions League triumph with Madrid, Ronaldo is the favourite to receive the trophy again this year to draw alongside his great rival.
Spanish media outlet Don Balon reported that Ronaldo "exploded" at the new offer from Madrid.
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Real Madrid are understood to be determined to keep the four-time Ballon d'Or victor, although Ronaldo earlier this month insisted that he did not want to sign a new deal.
However, he needs more love from the club. Ronaldo told beIN Sports in October that he is happy with the contract he has now.
"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.
Ronaldo has endured a hard start to the current campaign, with just one La Liga goal all season. He recently became a father once again after he and his girlfriend Georgina Rodriguez gave birth to their daughter Alana Martina.