But Adkins, whose side were undone against City by goals from Albert Adomah and David Clarkson, made clear his determination that his present club will not be left out of pocket should he move to St Mary's.
"I was not going to walk away from Scunthorpe while the question of compensation was unresolved," he added.
"I do know there was a big discrepancy between the amount that Southampton were offering and the figure that Scunthorpe were wanting.
"Southampton are a club now in the bottom four of League One and obviously in need of new leadership.
"It's a big club with big potential and anyone would be excited by the prospect of taking over there."
Adkins also emphasised that the question mark hanging over his Glanford Park future had no bearing on the loss to City.
"I made sure the players were focused and obviously this is not the result I wanted," he stated. "The first half was disappointing but I thought we battered Bristol in the second half without being able to get the goal we wanted."
City boss Keith Millen was delighted to collect his first Championship win of the season. "Hopefully this result will kick-start things for us now," he said.
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