Morgan took charge on a short-term basis a month ago following Danny Wilson's departure and the former Blades captain maintained the club's npower League One promotion challenge, which ended on Monday with defeat to Yeovil in the play-off semi-finals.
The 36-year-old informed chairman David Green of his desire to remain in the job before leaving Huish Park and believes resolving the situation as soon as possible will be best for all involved.
"I want to be the one who leads the lads and I've told the board and the chairman I want the job but it's not my decision," Morgan told reporters after the defeat in Somerset.
"They are the ones who make the decision and I'll probably speak to the chairman on the way home. He's excellent like that. Win, lose or draw he always calls and asks if there is anything he can do or if there is anything we want.
"I feel as though, despite what's happened here, in the three weeks or so we've been in that we've done well.
"It's probably best to have a quick decision so we can start planning straight away. That's got to start right now.
"I haven't made any firm plans in my mind because we've been preparing for two big games. But, yes, I do have thoughts in my mind of what this team and club needs."