The Gunners have insisted that Sanchez is not for sale this summer, with manager Arsene Wenger recently suggesting that the Chile international will see out the rest of his contract.
Less than 12 months remain on the attacker's deal, and there is no indication that the 28-year-old, who has also been linked to Bayern Munich and Paris Saint-Germain this summer, is close to signing a new one.
According to The Mirror, City are hopeful that Arsenal will agree to sell before Thursday's deadline if the two-time Premier League champions include Sterling in a deal.
The former Liverpool winger, who joined City in a £50m transfer in 2015, has scored two goals in three Premier League appearances so far this season.
It is believed that Pep Guardiola's club have now made direct contact with Arsenal regarding a possible deal for Sanchez, but the Gunners may want a fee of around £70m before they consider selling.