The Canaries stopper narrowly missed out on a place in Roy Hodgson's England squad this summer and a move to the Emirates Stadium would increase his chances of regaining his spot.
Arsene Wenger meanwhile is on the hunt for a 'keeper, having seen Lukasz Fabianski join Swansea City and Emiliano Viviano return to Italy following a loan spell
The Daily Mirror is reporting that Championship side Norwich will demand around £7million for the 27-year-old - significantly less than the supposed £15million that Cardiff City want for David Marshall.
Whilst the big-name signing of Alexis Sanchez earlier in the week has given reason for optimism, Arsenal fans will demand more signings as the squad looks in desperate need of several new players.
Along with a 'keeper, Wenger will also need to wrap up a deal for a right back - with Mathieu Debuchy the expected target - and many believe a defensive midfielder is also on the cards.
The likes of Sami Khedira and Lars Bender would have Gooners rushing to the club shop for their new Puma kit, but Frenchman Wenger knows competition for first-choice Szczesny is a priority.
With around a month before the league season kicks off the Pole is Arsenal's only senior 'keeper.
And should Arsenal's bid fail to tempt Norwich into a sale, the likes of David Ospina and Keylor Navas have also been mooted as possible options.
The South American pair shone at the World Cup, although Englishman Ruddy would have the advantage of knowing the league.