The caretaker boss has been unable to halt City's slide towards the Championship since replacing Chris Hughton last month, and Sunderland's midweek win against West Brom sealed their relegation in all but mathematics.
Adams has taken just one point from four games, but the former youth-team coach is in no doubt he has what it takes to manage the club full-time.
"I thoroughly enjoyed being involved with first-team football, so absolutely (I would like to stay on)," he said.
"I don't know what is going to happen, but it is something I am comfortable doing, I feel I am suited to it, and have the skill set to do it. I made the step up to first-team football and it is something you don't want to step back from.
"I have got my fingers crossed and I am hopeful that I am able to carry on in some capacity, hopefully as the manager, if that was not to be then obviously I would have to look at it as and when."
Malky Mackay, a former Norwich team-mate of Adams, has been linked with the manager's job after settling his dispute with former club Cardiff.
Asked about the prospect of forming a 'dream team' with Mackay, Adams said: "Is it a 'dream team'? I don't know. It is nice of you to say so, but I can't comment on that."
Adams described relegation as "bitterly disappointing", and said: "I feel the club deserves to be in the Premier League, it is where Norwich City should be playing their football.
"We are not in it, but hopefully in a year we will be back in there."
For now Norwich finish their season with a home game against Arsenal on Sunday, and Adams said: "There is everything at stake.
"There is a lot of money in terms of position within the Premier League and for me there are 27,000 reasons why there is everything at stake.
"Is it going to keep you in the division what happens on Sunday? No. But those fans deserve a performance."