The Blades will begin a fifth season in League One next month and Adkins is the fifth manager tasked with leading them back to the Championship following relegation in 2011.
Former defender Nick Montgomery, now in Australia with A-League side Central Coast Mariners after spending 13 years at Bramall Lane, said this week the Blades job was “the most pressurised in the country”.
But Adkins, unfazed after winning four previous promotions and three from League One, said in response: “Isn’t that brilliant? That’s what I want to be, I want to be in the Premier League.
“I would love it to be with Sheffield United. It’s one stage at a time and I’ve just said we’re very respectful of everybody in this division because I know what a grind it is.
“Is it the toughest job? It’s great. I want to manage at the top level and to do that you’ve got to manage in tough environments.
“It’s a really exciting one, a positive one that I’d like to think is being created at the moment and it’s a very exciting challenge for all of us.
“We should embrace that, that toughness, and bring it on because that’s what we want to be able to do.
“If you want to manage at the top level you’ve got to manage in tough situations, so I’m really excited about it.”
Adkins has not shied away from the club’s target of promotion since replacing Clough at the start of June.
He added: “We want to go up, no two-ways about it, we want to go up.
“But to do that we’ve got to be respectful of everybody else because if you underestimate them then you get a smack on the backside don’t you?”