The pressure is firmly on Arsene Wenger at present, after a disappointing campaign for his team will end without them winning anything.
Many feel they should have been title contenders this season as their rivals have preformed poorly, but instead they are now scrambling to secure a top four place.
As a result some Arsenal fans want him to end his long reign at the club and talk is rife who could replace him, with suggestions ranging from Diego Simeone to Dutchman Koeman.
And Savage believes that should the Gunners offer the Dutchman the position, it would be too good for him to turn down.
“If Wenger does leave ,why on earth would Koeman not want to take a job like at Arsenal when he’s at Southampton?” he told BBC Radio 5 Live 606.
“Don’t get me wrong, Southampton are a fantastic football club, but surely if he had the opportunity to manage Arsenal in the Premier League, that would be too good to turn down.”
Koeman would surely be one of several targets for Arsenal should Wenger decide to leave at any point.
While Simeone is the appointment that the fan base want given his success with Atletico Madrid, his methods would be an abrupt change in style for the club as a whole.
Koeman though, would be a more comfortable transition given he the style he has implemented at the Saints and that would surely put him among the contenders for the job, at least for the Arsenal hierarchy.