The winger was on loan at the Italian outfit during the 2009/10 season.
Mourinho lead Inter to the treble in that season, but Arnautovic only featured for the club three times.
However, the West Ham winger has revealed a particularly memorable moment from his year with the Portuguese boss.
"We were at Abu Dhabi for pre-season with Inter in 2009," Arnautovic told the Daily Mail.
"I overslept for breakfast. Late number one.
"Then there was the team meeting. The hotel was huge and my room was on the 75th floor.
"I am waiting for the lift and I press and I press and still I have got seven minutes so I am OK. But then I still didn't make it. Disaster number two.
"Then before we meet for the game it was happening again and I was like: 'He is going to kill me'.
"So then we fought a little bit and I was out of the team that day, training on my own."
After upsetting the outspoken boss, Arnautovic wanted to make it up to Mourinho by arriving at training early.
"Back in Milan I thought we were training in the morning and I went in just perfect," he continued.
"There were no cars there. We were actually training in the afternoon that day!
"Mourinho is there with his staff and he stands up and starts applauding and laughing.
"He said, 'You are my man. You come here five hours before training. I love you! Here, take my watch'.
"I still have that watch in my house.'"
The Austrian has struggled to live up to his £27m price tag so far this season but admits he is working hard to put things right at the London Stadium.
"Expectation of me is high and I know people are not happy,' Arnautovic said.
"I have apologised for the red card, it was stupid. And then I've been ill and I can't change it.
"So it's not been great and I know that.
"I hear people saying I came for £27m and have done nothing. I want to show the club they didn't pay so much money for nothing."
