Dzeko has failed to make an impact at City since joining the club from Wolfsburg in January.
The Sun reports that Real Madrid boss Jose Mourinho is ready to make a £20million bid for the striker who joined City for £27million in January.
Dzeko signed a four-and-a-half-year deal when he joined City in January.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.