The Gunners were one of several sides showing an interest in former Leeds United attacker Raphinha alongside Chelsea and Tottenham Hotspur.
There was optimism of the Brazil international remaining in the Premier League, with Barcelona initially struggling to stump up Leeds' asking price.
However, Raphinha had his heart set on a move to Barcelona, who have now confirmed the attacker's arrival at Camp Nou after a protracted transfer saga.
Arsenal are still said to be in the market for a new wide man before the transfer window shuts, with new signing Marquinhos potentially heading out on loan for the 2022-23 season.
According to journalist Ben Jacobs, there have been 'rumblings' surrounding a possible move for Sane from Arsenal, with Jacobs telling The Done Deal show: "A move to La Liga could be more likely.
"I've not heard rumblings at the Chelsea end, I've actually heard more rumblings from Arsenal that this was a name being considered at the beginning of the window, but then they turned their attentions to Raphinha."
Real Madrid have also been tipped to swoop for Sane, who would supposedly command a £59m figure to leave the Allianz Arena in the current window.
However, Blancos boss Carlo Ancelotti does not expect his side to make any more signings after bringing Antonio Rudiger and Aurelien Tchouameni to the club.
Sane worked with Arsenal boss Mikel Arteta during his time at Manchester City, and the 26-year-old has hailed the Spaniard's work at the Emirates so far.
"I can just say positive things about Mikel Arteta. Since we started working together, he helped me a lot on the pitch," The Mirror quotes Sane as saying.
"He tried to improve my football; he told me what I did good and what I did wrong, and he tried to work on it. Next to Pep [Guardiola], they both really helped me and improved me, to bring me to the next level.
"Even now, at Arsenal, for me you can see he's doing a great job. You can see he has a concept, a way that he wants to play, a way that he wants his team to play, and obviously it will take some time until everybody really understands it. I think he will do a great job as a [head] coach too."
Sane left Man City to sign for Bayern in the summer of 2020, and the Germany international has since registered 24 goals and 27 assists in 89 games for Julian Nagelsmann's side.
The attacker still has three years left to run on his deal at the Allianz Arena, but Bayern may be reluctant to let another attacker leave after sanctioning Robert Lewandowski's exit for Barcelona.
The Gunners were also briefly tipped to bring Serge Gnabry back to the club from Bavaria, but the ex-Arsenal youngster has instead penned a new long-term deal with Bayern until 2026.