According to The Express, the 29-year-old Portugal international is thought to be unhappy with the sale of Angel di Maria to Manchester United and Xabi Alonso to Bayern Munich respectively and is "determined" to leave the Spanish capital.
Los Blancos are rumoured to be looking for a fee in the region of £60m for the 2014 FIFA Ballon D'or winner and the Premier League champions are said to be a viable destination.
However, City may face competition from Ligue 1 champions Paris Saint-Germain, with Les Parisiens willing to meet his wage demands and Real's price tag.
The possibility of a return to Old Trafford remains, but manager Louis van Gaal is reportedly mulling alternative options.
Ronaldo scored a penalty in Real's 2-1 derby defeat at the hands of Atletico Madrid on Saturday.