The Citizens have admitted defeat in their attempts to sign the Chile international before the summer transfer window closes at 11pm on Thursday night.
According to The Mirror, however, the long-running pursuit is not over yet as City will table a third bid when the transfer window reopens for business in four months' time.
City are said to have been confident of getting a deal over the line on deadline day, only for Arsenal to fail in their pursuit of landing replacement attacker Thomas Lemar, who instead opted to stay at AS Monaco.
A source is quoted as saying: "It was Ivan Gazidis who made the first move this morning – calling Ferran to tell him that the second offer was acceptable. It came with the proviso that Arsenal had to sign a replacement. But Gazidis indicated that was close to happening.
"It now seems that Arsenal didn't have an agreement with either a club or a player. It's disappointing but we hope that the strong relationships we built with Alexis and his advisers will help us in the future.
"Arsenal have known for four months that we were interested in signing Sanchez and that the player wanted to leave the club."
Sanchez played a direct part in 34 Premier League goals last season, but a minor injury problem has restricted him to just one outing so far this term.