Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Adkins, who was dismissed by Saints in January after a three-year stint at St Mary's, is expected to take over at the Madejski Stadium ahead of Saturday's Premier League match at Arsenal.
The Royals sacked Brian McDermott two weeks ago - just 33 days after he was named manager of the month - with Academy director Eamonn Dolan taking charge.
With just eight league games remaining, Reading face a desperate fight against relegation - in 19th place and seven points adrift of safety.
Former Swindon boss Paolo Di Canio had been linked to the post, as well as Roberto Di Matteo, while Brighton's Gus Poyet is said to have turned down the job last week.
Adkins has also been linked with Leeds United and Leicester City in recent weeks.
Bookmakers saw a plunge on Adkins during the day on Monday but Reading have yet to comment.