The Daily Mirror's John Cross says City have tabled the offer in a bid to sign Sanchez before the transfer window closes on Thursday.
The BBC said should City's offer be more tempting, they might sell if City would also trade Raheem Sterling along with the transaction.
Sanchez has reportedly asked for permission to leave the Chile worldwide squad for 24 hours amid latest reports of a transfer bid from Manchester City.
Manchester City have been looking to secure a deal for Sanchez this summer.
The former Udinese star has already worked with Pep Guardiola at Barcelona and is reportedly willing to reunite with his former boss. City are the only realistic buyers left for the striker after both Bayern Munich and Paris St Germain dropped out once Sanchez's wage demands were revealed.
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Arsenal have suffered a poor start to the season, with an unconvincing opening day win over Leicester City followed by a 1-0 loss to Stoke and a 4-0 thrashing at the hands of Liverpool on Sunday.
Raheem Sterling is one of several options that has been mooted during negotations between the two clubs in the past 48 hours, though Goal understands it is the Gunners who have pushed for his inclusion.
England forward Sterling, 22, has featured in all three of City's games so far this season.