The Mirror says the Spanish club are understood to want Tevez, whose City career is effectively over following his alleged refusal to play against Bayern Munich in the Champions League last month.
City will this week announce what action they intend to take against him following an internal investigation into the striker's Munich mutiny.
Tevez, whose two-week suspension ends on Thursday and who is back home in Buenos Aires, is wanted by Real boss Jose Mourinho. But a potential January deal could be scuppered by the size of the fee and the former City skipper's wage demands.
City want at least £35million for Tevez, but the striker will have to overcome the stumbling block of Real being unwilling to meet his wage demands of £250,000-a-week.
Cristiano Ronaldo is Real's top earner on £200,000-a-week.