The Switzerland international has made a name for himself with Die Adler since joining from BSC Young Boys in 2019, having previously represented Borussia Monchengladbach's youth academy.
Sow - who can operate as a number six or number eight - has produced eight goals and 13 assists in 160 games for Frankfurt, winning the Europa League in the 2021-22 campaign.
The 2022-23 season was another productive year for the 26-year-old, who netted four goals in 47 matches across all tournaments, but he has now entered the final 12 months of his contract.
A recent report claimed that Arsenal were pushing hard to bring Sow to the Emirates during the current window, as his contract situation offered hope of a cut-price deal.
However, transfer expert Fabrizio Romano claims that Sow has instead chosen to make the move to Sevilla and has given the green light to a five-year deal until the summer of 2028.
Despite Arsenal's apparent attempts to hijack the move, Sow is on the brink of joining Sevilla for approximately €14m (£12m), and he completed his medical tests on Wednesday.
While the midfielder's decision comes as a soft blow to Arsenal, the Gunners have already splashed out £170m to reinforce their engine room with the double capture of Declan Rice and Kai Havertz.
The pair arrived in North London to replace Bayer Leverkusen-bound Granit Xhaka, while Thomas Partey has also been linked with a move away from the club following Rice's signing.
Arsenal could have sought out another lynchpin had Partey completed a move to Juventus or Saudi Arabia, but the Ghana international is now expected to continue in red and white for at least one more season.
With the signing of Jurrien Timber taking Arsenal's spending to over £200m, the Gunners will likely be focused on outgoings for the remainder of the window, but a move for Brentford number one David Raya is still in the works.
The Gunners made their third sale of the summer window on Thursday, as American centre-back Auston Trusty joined Premier League new boys Sheffield United for a reported £5m.
As well as Xhaka and Trusty, Pablo Mari's move to Monza was made permanent, while Ainsley Maitland-Niles left the club upon the expiration of his contract in June.
The likes of Nicolas Pepe, Albert Sambi Lokonga, Nuno Tavares, Alex Runarsson and Cedric Soares are expected to be the next names out of the door, and they could be followed by Arthur Okonkwo, Rob Holding, Folarin Balogun and Matt Turner.
Arsenal retained the Emirates Cup with a penalty-shootout win over Monaco on Wednesday and are now gearing up for Sunday's Community Shield clash against Manchester City.