Morgan, 37, will coach the Under-21s under Adkins, who replaced the sacked Nigel Clough as the Blades’ eighth manager in as many years at the start of the month.
“I interviewed for the job when Danny (Wilson) left before David Weir got the job and I was unsuccessful at that point and probably through eagerness I threw myself at it,” Morgan said.
“I genuinely wanted the job but David got it and I think at that point it gave me an opportunity to sit down and analyse and I looked at how unfortunate David was with his brief tenure.
“I thought that could very easily have been me so I’m under no illusions. It’s not an easy job because I’ve seen that, working with vastly experienced managers like Danny Wilson and Nigel Clough.
“I’m not 38 while November. Management is something I want, it’s something I want to go into.
“I completed my UEFA Pro licence last October, I feel like I’ve gained valuable experience as a coach and a player.
“So when the opportunity does come up, which I’m sure it will. Nigel Adkins asked me if I wanted to manage one day and I said I did.
“But I’m genuinely not in a mad rush and I’m looking forward now to seeing how Nigel and his staff work and learning a bit more from them.”
Morgan’s testimonial match against Newcastle takes place on Sunday, July 26.
The former Blades skipper, signed from hometown club Barnsley in 2003, led United to promotion to the Premier League in 2006.
He made 247 appearances for the Blades before taking on his first coaching role in October 2010.