Jose Mourinho gave talkSPORT a hilarious tale about his first encounter with the Austrian forward when in charge of Inter Milan.
Inter snapped up Arnautovic on loan from Dutch side Twente in August 2009 but Mourinho says they had been tracking him for some time.
When they began their search, Mourinho had the likes of Zlatan Ibrahimovic, Adriano, Hernan Crespo and Julio Cruz to choose from.
But Arnautovic made a stunning declaration when meeting the Portuguese at Inter's training ground: "I'm better than all of them."
Mourinho chuckled when asked about the former West Ham man on Tuesday's talkSPORT Breakfast.
The Special One recalled: "My technical director at Inter, Marco Branca, he arrived with that player that he followed and he wanted to bring over from Holland. He [Arnautovic] was 18.
"When Branca arrives with him we were training. They stay there on the touchline and in the middle of the training session I decided to go over and say hello to them.
"At the time, in my first season at Inter, my strikers were Ibrahimovic, Adriano, Hernan Crespo and Julio Cruz.
"I go to Branca and Arnautovic, we shake hands and share a few words and Marko tells me: 'Mr, I'm better than all of them.'
"Ibrahimovic, Adriano, Hernan Crespo and Julio Cruz - Marko is 18 and he's better than all of them!"
Mourinho confirmed Arnautovic has always possessed 'incredible confidence' but that kind of attitude makes him an acquired taste.
Arnautovic, now 32 years old, landed himself in trouble during Austria's first match of Euro 2020 as he was ruled to have used insulting language towards North Macedonia players.
He was given a one-match ban for the offence.
But Mourinho believes the player is somewhat misunderstood, saying Arnautovic is different off the pitch than he is on it.
He added: "This is Marko. A kid with incredible confidence, incredible happiness, very emotional, very sweet, he's everything that you want in a group.
"The group loves him, he can have these bad emotional reactions but he was an amazing kid and now he's an amazing family guy.
"He's completely different in terms of his life outside the professional life, but this is Marko."