Southgate is under pressure after their 1-0 defeat at Manchester City on Saturday made it 13 league games without a win for the club.
He said: “There is no chance that I will walk away from the club. I won’t shirk from the challenge. I have never walked away from anything during my career as a player or a manager.
“It is for me to carry all the pressure. All the chances fell to Afonso Alves today but it isn’t as through he missed them.
“Shay Given has denied him with some fantastic saves. We made one mistake all game and we were punished for it.”