The 58-year-old stepped in for the Canaries' 2-2 draw with Blackburn Rovers on Saturday afternoon, less than 24 hours after Neil was given the shove.
Irvine was brought to Carrow Road by Neil in July to provide an experienced head in the dugout, but City's hopes of a first-time return to the Premier League have since evaporated following a poor run of form.
"I got a phone call late on Friday afternoon, just as I thought I was going to have a quiet Friday night, and I was simply asked to take the team for this game and the next one," he told reporters. "When you are asked by the board to do that, that is what you do so I agreed to that. Alex came back to the training ground after the board meeting. It was good to see him.
"I won't apply for the job. I told Alex right from the beginning I am not a threat to him. With my age and experience that could have been regarded as such when I first came in. Even before I agreed to come I wanted that made clear. I can't tell him that and then apply for the job. What my role is in the future is not in my hands.
"I may still be at the club in the same capacity, or a different one or I could be away. This is a great club. I made a decision to move 250 miles away from home in the eagerness to get back to the Premier League. I think this club is capable of getting back. It will be difficult but this club should be aiming for that, if not this season then next. I don't regret my decision and if the club want me to be here I would like to be here."