New City boss Manuel Pellegrini is keen to bring in a new centre back, as Matija Nastasic will be out of action for the first month of the new season with an ankle injury.
City had a £15million bid for Pepe rejected last month and Real Madrid are keen to keep the Portugal international.
The Daily Mail reports that City are set to make a final attempt at signing Pepe before the start of the new season.
City plan to make a bid of around £20million and director of football Txiki Begiristain has flown to Madrid for talks with the Spanish side.
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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.