Southgate said: "I would like to place on record my thanks to Malcolm Crosby for his work and support over the last three years.
"Without doubt, this has been the most difficult decision I have taken because Malcolm is a true gentleman and a real football man.
"My decision has been based on re-organising the way my backroom team operates next season.
"I hope and am certain that Malcolm will find a job in football soon. I wish Malcolm and his family the very best for the future."