The Sun says City boss Pep Guardiola was confident the 28-year-old wanted to join them, despite a failed £60m pursuit in the previous transfer window.
But Paris Saint-Germain are ramping up their interest and want him to partner Neymar and Kylian Mbappe upfront.
Sanchez met with PSG chiefs in the French capital this summer, where they discussed a potential transfer.
He demanded a staggering £400,000-a-week, which the Ligue 1 side admitted they could not afford to pay.
But Unai Emery's team did tell him they would hand him a hefty signing on fee and top City's offer of £275,000-a-week.