The Sunday Mirror says Adkins is the bookmakers' favourite to be the first Premier League boss to go after the Saints began their return to the top flight with three straight defeats.
Redknapp would not be a popular appointment on the South Coast after his ill-fated previous spell with the Saints, but another man on the list of out-of-work bosses could well catch chairman Nicola Cortese's eye.
Benitez has been has been out of management since being dispensed with by Inter Milan in December 2010.
But the Spaniard is a Champions League winner with Liverpool, and also guided the Anfield club to an FA Cup success. He was surprised not to be approached to make a Liverpool return after Kenny Dalglish was sacked.