Arnesen, who has previously held the position at Tottenham, Chelsea, PSV and Hamburg, believes the former Arsenal midfielder has a skill set that makes him perfectly suited for the position and is potentially ready to move up if required.
Vieira is head of City's Elite Development Squad, having also spent two years prior to that as football development executive at the Etihad Stadium, all vital experience that Arnesen thinks prepares the Frenchman for the next step in his career, whether that is coaching or a technical role.
"Vieira's going the right way, he's not straining for it. They [City] should maybe look at him," said the Dane ahead of his appearance at a sporting directors' summit at Manchester Metropolitan University in April.
"He's super-intelligent, a very good guy, speaks languages, which is important these days in modern football when you get players from all over the place.
"Vieira has to make a decision. Do I go the coaching way? Do I want to be head coach or do I want to be technical director?
"He's now getting ready to do a senior job, where I don't know.
"If you had put him in straightaway [directly after he finished playing], you would have lost him because he wouldn't know what to do. That would be too dangerous."
While Arnesen reckons Vieira could feasibly replace Begiristain, he has expressed surprise at suggestions from Dimitri Seluk, Yaya Toure's agent, tipping the Ivorian to be jettisoned into a prominent position behind the scenes at City.
Seluk claimed Toure wants to play out his days with the Premier League champions and "when he stops playing, then Yaya could have an executive role, perhaps as sporting director".
But Arnesen said: "It's impossible [to go direct from player to director of football].
"I always think it is great to have ex-players going in to work for the club, but you will kill the player if you get him in without any experience on the coaching side.
"It's not fair towards the player or towards the club.
"Many times you see a big, big player and they think we have to give him the possibility, but it's not giving him a possibility. You will kill him. In the long run you will lose maybe a potentially fantastic one who can become a great sporting director."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.