The 25-year-old custodian has been identified by the Gunners as an ideal replacement for Emiliano Martinez, who sealed a move to Aston Villa on Wednesday.
Arsenal have been in talks with French Ligue 1 club Dijon over a deal for the Iceland international and are claimed to have agreed a fee in the region of £1.5m for the player.
Runarsson is now scheduled to undergo a medical with north London club today ahead of his move from Dijon to Arsenal.
The goalkeeper will come in as back-up to Arsenal number 1 Bernd Leno.
Gunners goalkeeping coach Inaki Cana worked with Runarsson during his time at Danish club Nordsjaelland and is said to be a fan of the Iceland star.
Mikel Arteta’s side also targeted Brentford goalkeeper David Raya, who also worked with Cana, as a replacement for Martinez.
However, the Bees have been adamant that the Spaniard is not for sale, having already sold Ollie Watkins to Aston Villa for top dollar; Arsenal remain interested though.