The Football Association and Premier League will investigate claims made by the BBC's Panorama programme.
Allardyce has been implicated by the Panorama programme
The documentary gathered information into the alleged payment of bungs and illegal approaches in the game.
"We expect this information to be provided as a matter of urgency," said a FA statement.
Bolton have also revealed they will launch a "thorough and robust" investigation into allegations which involved boss Sam Allardyce.
The FA statement added: "The FA takes any allegations of corruption in the game extremely seriously and will actively investigate the claims made by the programme."
"Some of these investigations will be conducted jointly with the Premier League when the claims relate to both FA and Premier League rules.
"Discussions have already taken place with the Premier League as to how these joint investigations will be managed."
WHAT IS A BUNG?
A secret and unauthorised payment seen as a financial incentive to help a transfer go through
HOW DOES IT WORK?
An agent pays a club official a slice of his cut for the help of the club official helping the transfer go through
An agent may... never so approach a player who is under contract with a club with the aim of persuading him to terminate his contract prematurely or to flout the rights and duties stipulated in the contract
Several leading Premiership clubs and managers as well as agents were implicated in the documentary.
FA chief executive Brian Barwick said: "These are serious allegations which have been made by Panorama and we are determined to investigate them fully.
"It is vital for the integrity of the game and for every football supporter that we do this.
"We will work in close co-operation with the Premier League. As with any investigation, we will ensure that our inquiries are exhaustive and thorough. If we find evidence of corruption we will act on it. We recognise our responsibility."
BBC Panorama has said in a statement that it will cooperate with requests relating to the programme from the relevant investigating authorities, and "will do so in line with our normal processes and procedures in cases of this kind."
Allardyce is at the centre of the allegations of bungs with footage showing two agents claiming they paid bungs to the Bolton boss, who denies ever asking for or taking a bung.
Agent Teni Yerima claims he bribed Allardyce in the past. Another agent, Peter Harrison, describes offering to pay the Bolton manager's son Craig, who Panorama claim received secret payments from agents during three transfer deals.
These allegations damage the integrity of football and need to be looked at properly
Sports minister Richard Caborn
Bolton issued a statement on Wednesday in which the club said they were taking "the allegations broadcast very seriously" but also vowed "not to over-react".
Bolton say they will launch "a thorough and robust investigation into the allegations".
Former England managerial candidate Allardyce also told the BBC he would not condone any breaches of FA rules - whatever personal affection he has for his son.
Craig Allardyce said he was exaggerating his own importance to the undercover reporter in order to attract opportunities.
He denied any wrongdoing in his Bolton deals or in his relationship with the club.
Portsmouth boss Harry Redknapp is secretly filmed talking about Blackburn defender Andy Todd.
Redknapp denies his conversation about Todd with Harrison amounted to "tapping up".
Redknapp's assistant at Portsmouth at the time of the filming, Kevin Bond, is secretly recorded admitting he would consider discussing receiving payments from a proposed new agency involving Harrison.
Bond also says he will discuss it with his then boss Redknapp.
Redknapp told the BBC that he has never taken a bung and had given Bond no reason to think otherwise.
When Bond was told of the filming he said that he is not interested in receiving bungs and that no one he has ever worked with has taken a bung.
The documentary also shows Chelsea's director of youth football, Frank Arnesen, secretly filmed making a illegal approach for or "tapping up" Middlesbrough's England youth star 15-year-old Nathan Porritt.
Chelsea have denied the filmed meetings broke any industry rules.
The Premiership champions were given a suspended three-point deduction by the Premier League when they were found guilty of illegally approaching Arsenal's Ashley Cole, who has since joined Chelsea.
There are allegations concerning other potential breaches of industry rules in the programme that fall outside the terms of reference of Lord Stevens' inquiry
But the Press Association has reported that Panorama's claims will not lead to that points deduction being triggered.
Chelsea are also currently the subject of a joint investigation by the Premier League and FA over claims by Leeds of illegal approaches for three academy players last season.
The Press Association claims the suspended sentence refers to 'contracted players', and not academy players.
Porritt was also the subject of filmed discussions between Harrison and representatives of Liverpool FC with Newcastle also implicated.
Panorama claim 18 past and present Premiership managers had been named as having taken illegal payments.
Sports minister Richard Caborn called on the programme to hand its evidence to the Premier League's inquiry, headed by Lord Stevens, which is due to announce its findings into illegal payments in football on 2 October.
"The integrity of sport needs to be upheld and there are proper rules for managers and agents. These allegations damage the integrity of football and need to be looked at properly.
"The programme alleged they had names of 18 managers who had received illegal payments, and I think they should give all their findings over to the Lord Stevens inquiry.
"This reinforces what I have been trying to do to bring in greater regulation into football through the European Football Review."