Allardyce was targeted in a Daily Telegraph investigation into alleged corruption in English football, and the departure of the 61-year-old was confirmed following crisis talks involving FA chairman Greg Clarke and chief executive Martin Glenn.
Former Bolton and West Ham manager Allardyce said: “Further to recent events, The FA and I have mutually agreed to part company.
“It was a great honour for me to be appointed back in July and I am deeply disappointed at this outcome.
“This afternoon (Tuesday), I met with Greg Clarke and Martin Glenn and offered a sincere and wholehearted apology for my actions.
“Although it was made clear during the recorded conversations that any proposed arrangements would need The FA’s full approval, I recognise I made some comments which have caused embarrassment.
“As part of today’s meeting, I was asked to clarify what I said and the context in which the conversations took place. I have co-operated fully in this regard.
“I also regret my comments with regard to other individuals.”