The Gunners have been pursuing a long term replacement for Petr Cech since the summer, but their attempts to sign Jordan Pickford were scuppered when Everton snapped up the Sunderland keeper for £30m.
Butland is seen as a good alternative and at 24 years of age he has a long career ahead of him, but the England international could cost £40m.
Liverpool are also believed to be pursuing the stopper, with Reds boss Jurgen Klopp said to be eyeing a new number one keeper after both Loris Karius and Simon Mignolet failed to stake their claim for the position.
Arsenal have been linked with a number of keepers over the past few months. Athletic Bilbao’s Kepa Arrizabalaga was heavily linked with a move to the Emirates before he signed a new contract in January, while Roma’s Brazilian keeper Alisson is also said to be on the club’s radar.