The Gunners are eyeing a new back-up stopper with Alex Runarsson is nearing an exit to Turkish Super League side Altay Spor.
It is now reported that the North Londoners have made an approach to sign Johnstone, who is likely to depart the Hawthorns.
The 28-year-old stopper has now entered the final year of his contract, and he could be available for £10m.
West Ham United have also expressed an interest in former Manchester United goalkeeper, but they are only willing to pay £5m up front.
Mikel Arteta’s side are willing to meet Johnstone’s price tag, but the England international would have to settle for a back-up role.
Bernd Leno has two years left on his contract at the Emirates.