The Gunners were widely fancied to sign the shot-stopper last month, but they have yet to make an approach.
Onana is currently banned from playing until November 4 while he will be on African Cup of Nations duty between January and February.
Owing to this, the Gunners are having second thoughts over a move, and this could pave the way for him to join Lyon this summer.
Les Gones have opened negotiations with Ajax, and they are currently negotiating the transfer fee for the Cameroon international.
The club are prepared to guarantee him a starting role after the ban. He also has the assurance of playing European football.
The Gunners could offer him neither. It is unclear whether Mikel Arteta would drop Bernd Leno midway through the season.
The club are looking at alternative targets including West Bromwich Albion's Sam Johnstone and Sheffield United's Aaron Ramsdale.
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