The Swans let Paulo Sousa leave the club on Sunday after he made it clear he wanted to replace Nigel Pearson at Leicester City. Pearson himself had only just quit the Walkers Stadium to assume control at Hull City.
Huw Jenkins, the Swansea chairman, has said he would like a young British manager - Sousa is Portuguese and his predecessor was the Spaniard Roberto Martinez.
He has identified Adkins, 45, as one the of the most promising young managers in the Football League, along with Rochdale's Keith Hill, Bournemouth's Eddie Howe, Yeovil's Terry Skiverton and Paul Trollope at Bristol Rovers.
"Nigel Adkins has also worked miracles at Scunthorpe and has done a very good job and is very loyal to the club," he told BBC Wales, suggesting Adkins should not be considered one of the favourites to be approached.
Adkins has been in charge at Glanford Park since November 2006, when he stepped up from physio when Brian Laws and his assistant Russ Wilcox defected to Sheffield Wednesday.
He led the Iron to the League One title in his first season, and although Scunthorpe were relegated after just one year in the Championship they won an instant return by winning the 2009 League One play-offs.
Adkins then managed to avoid relegation last season despite operating on the lowest budget in the division.
That record saw him installed as on of the early favourites to replace Sousa, though he has now drifted away in the odds as itbecame clear he is unlikely to leave the Iron for Swansea.
As of 2pm on Friday he was available with Skybet at odds of 10/1. Gary Speed and Gianfranco Zola are joint favourites at 6/4.
Speed was interviewed for the job last summer, wen Jenkins opted for Sousa.
Steve Wharton, the Scunthorpe chairman, told the Scunthorpe Telegraph on Friday that there had been no approach from Swansea for Adkins.
Adkins does have experience managing in Wales, having coached Bangor City to the League of Wales title in 1995 and 1996 as player-manager.