The 63-year-old left his post by mutual consent after just 67 days in September 2016 following a Daily Telegraph investigation into corruption in football.
In a statement released on Wednesday evening, Allardyce hit out after the Independent Press Standards Organisation upheld in part a complaint over claims he had offered advice on how to circumvent FA and Fifa rules on third party ownership of players.
Allardyce said: "Had the FA stuck to their word and waited to see the Telegraph's evidence (as they originally told me they would), they would have seen that the allegations made against me were false.
"It was of course the allegations about third party ownership that the FA stated were the reasons for my leaving.
"It was clear that those that I was dealing with were more concerned with their own image than getting to the truth of what had occurred.
"I will consider my position in this regard with my lawyers."
Allardyce took charge of the national team for just a single game, a 1-0 World Cup qualifying victory in Slovakia, but was handed a route back into football in December 2016 by Crystal Palace.
He said: "I waited many years to be England manager and believe I would have made a success of the position.
"It took me a long while to recover from the disappointment of losing the position I'd worked so hard for because of stories that I knew were false.
"I was very lucky that first Crystal Palace and then Everton gave me the chance to rebuild my reputation and enjoy the game again.
"I wish Gareth Southgate all the best with the England position and he has shown that he is more than capable.
"There are some great players in the England set-up and I hope they are not let down by the administrators."
The Telegraph acknowledged the IPSO decision in a correction printed in Thursday's edition of the paper.
It accepted that Allardyce, while suggesting a way that a third party could share in the financial rewards of a transfer, had made clear that a third party could not take a portion of the transfer fee.
The Telegraph also accepted that Allardyce had not entered into negotiations to be paid to give advice on third party ownership, and admitted "an inaccurate claim was made that Mr Allardyce had briefed on 'breaking the rules'."
The correction said: "On these three points alone, the articles were inaccurate.
"This correction has been published following a complaint upheld in part by the Independent Press Standards Organisation."
An article in Thursday's edition of the Telegraph also stated: "On the major issues raised by Mr Allardyce, IPSO found for the Telegraph.
"It upheld our right to use subterfuge and secret filming, since there was a strong public interest in investigating and it was reasonable for the Telegraph to have believed that it could only obtain material evidence through subterfuge."
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