It follows a newspaper investigation claiming he offered advice on how to "get around" rules on player transfers, the BBC reported.
Allardyce, 61, is also alleged to have used his role to negotiate a deal worth £400,000 to represent a Far East firm.
An FA statement said Allardyce's conduct "was inappropriate" and Gareth Southgate will take temporary charge.
"He accepts he made a significant error of judgement and has apologised," the FA said.
"This is not a decision that was taken lightly but the FA's priority is to protect the wider interests of the game and maintain the highest standards of conduct in football."
"The manager of the England men's senior team is a position which must demonstrate strong leadership and show respect for the integrity of the game at all times."
Allardyce succeeded Roy Hodgson in July following England's disastrous performance at Euro 2016 in France and becomes the national side's shortest-serving full-time manager, the BBC added.
The Daily Telegraph had said Allardyce had a meeting with undercover reporters posing as businessmen before he took charge of his first England training session.
During the meeting, which was recorded on camera, it is alleged Allardyce said it was "not a problem" to bypass rules on third-party player ownership and claimed he knew of agents who were "doing it all the time".
It was further alleged that a £400,000 deal was struck for him to represent the company to Far East investors and to be a keynote speaker at events.
In the meeting, Allardyce also referred to Hodgson as "Woy", making fun of his predecessor's manner of speaking, and criticised Gary Neville, one of Hodgson's assistants.
Allardyce met FA chairman Greg Clarke and chief executive Martin Glenn on Tuesday to offer what he called a "sincere and wholehearted apology for my actions".
Clarke told the BBC he had not been sacked, but they had agreed his position was "untenable".