The 48-year-old was dismissed from St Mary's following two-and-a-half years, during which Adkins led the side from League One, through the Championship and to the top flight of English football.
"Situations happen in life and in football," Adkins said in a press conference. "I had a fantastic relationship with the supporters and everyone at Southampton but that has now gone.
"We had a fantastic journey there and that can never be taken away. We were with the players for two-and-a-half years and we were very successful in that spell and we played some good football."
Adkins will face his former club for his second game in charge at Reading on Saturday.