The Telegraph reported on Sunday that Arsenal were in talks with Dijon over a £1.5m deal to sign the 25-year-old stopper after he was recommended to Arteta by goalkeeping coach Inaki Cana - who worked with Runarsson during their time together at Nordsjaelland.
However, with Matt Macey also being linked with a move away from the Emirates, there is a chance Arsenal are bringing in Runarsson as a third-choice goalkeeper and are still on the hunt for a Martinez replacement.
"Signing for this special club was never in doubt", he said.
"I believe in Arsenal".
Arsenal head coach Mikel Arteta now has a very big squad that will have a lot of players that will want to take part in matches and we today take a look at two players that will give him squad choosing headache when they become fit to play.
Whilst the signing of a backup keeper mightn't excite the fanbase too much, Arsenal and Arteta know all too well the importance of a reliable understudy, having watched Martinez come in for the final games of the season and flourish.
Gunners boss Mikel Arteta could have found his replacement immediately with the four-times capped Iceland worldwide on the radar and available for what could be a very low fee. He's a superb player with an incredible mentality.
"He's an important leader for the team and a big part of what we're building".
Gabriel Martinelli is one of the best prospects in the world as he scored 10 goals last season as well as making two assists but he however unfortunately got injured in training post lockdown a few months ago. "He can achieve that here". "It's a big boost for everyone - fans, team-mates and staff".