He has agreed terms with the Saints to takeover at St Mary's, along with assistant Andy Crosby.
The two clubs have yet to agree on a fee for compensation, with the Iron said to be asking for around £200,000. Adkins was in charge at Glanford Park on Saturday as the Iron were beaten 2-0 by Bristol City.
"If it is my last game it is very disappointing to lose, especially when we've played so well in the second-half. If it is the final game then all I can do is thank a fantastic group of players, support staff and supporters who have been fantastic," he said.
"I do know there was a big discrepancy between the amount that Southampton were offering and the figure that Scunthorpe were wanting.
"It is a great opportunity to go and take over Southampton, I was offered the job and a decision has got to be made. Hopefully that will be sorted Saturday evening or Sunday at the latest because Scunthorpe have got an important game at Sheffield United on Tuesday night.
"They are in a division lower than us and there has been uncertainty behind the scenes, but I have thought long and hard about it. A conclusion needs to be drawn now so everyone can move on. Scunthorpe United have got to get ready for Tuesday's game."
Adkins praised his players, saying they had not been affected by the doubts over his future.
"I thought the second-half performance, the way we've dominated the game and created so many chances was very entertaining for our supporters," he added.
"In the first 15 minutes of the first-half we did okay, Adomah agot goal and that changed the complexion of the game. At half-time we sat down and said we could improve on our performance, and great credit should go to the players.
"It was a 50 minute battering for Bristol City. We've played some really good football, with lots of movement and endeavor."