A compensation agreement between the two clubs had not been reached on Thursday morning, but their move to the St Mary's Stadium is now inevitable.
A move would bring to an end a week of speculation, which quickly saw Adkins installed as one of the favourites with bookmakers.
The Saints asked for permission to talk to Adkins on Tuesday night, with talks taking place over compensation and personal terms on Wednesday.
Adkins had been in charge at Glanford Park since November 2006, when Brian Laws left to manage Sheffield Wednesday.
He led the Iron to promotion from League One twice, and avoided relegation from the Championship last season. He was in charge for 198 games, winning 83, drawing 44 and losing 71.
Ian Baraclough looks likely to be in charge of the Iron against Bristol City on Saturday.