Manchester United manager Louis van Gaal has refused to rule out the possibility of signing goalkeeper Victor Valdes.
Manchester City boss Manuel Pellegrini says he is confident that England goalkeeper Joe Hart will sign a new contract.
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.