The seven UK betting companies with exclusive rights to screen FA Cup matches have offered to cancel the exclusivity of the deals and allow the Football Association to show games itself, following sustained criticism from campaigners and politicians.

The offer, made by the Betting and Gaming Council on behalf of the companies – Bet365, Betfair, William Hill, Coral, Ladbrokes, Unibet and Paddy Power – was announced after the FA had begun talks with the marketing giant IMG about the possibility and likely cost of withdrawing from the exclusive element of the controversial arrangement.

IMG bought the FA Cup media rights in early 2017 then sold to the seven gambling firms the deals under which they could exclusively live-stream on their websites matches that did not kick off at 3pm on a Saturday and were not picked for live broadcast by the BBC or BT Sport.

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The FA decided in May 2017 that it would cancel its then partnership with Ladbrokes and not accept any sponsorships from betting sites, but the rights deals for the FA Cup with the betting companies had already been concluded by IMG, from 2018 to 2024. Given the FA’s public stance, when the outcry over the deals erupted this week, officials moved immediately to see whether the governing body could withdraw from them, although they feared the betting companies could insist on very substantial compensation.

A senior gambling industry source told the Guardian that the betting companies would not be looking for any refund from IMG or the FA for giving up the exclusive broadcast rights, and would write off the cost of the deals.

The sports minister, Nigel Adams, had called on the FA to review the deals, following a storm of criticism that such deals tie football support to gambling much too closely. Instead, the betting companies themselves volunteered to give up that key element of their rights, to show the matches exclusively to people who had already placed a bet on their site or had an account with money in it.

The FA itself had insisted on that condition of the deals; the governing body is understood to have considered it a protection against general gambling marketing to football supporters, restricting the live streaming to existing betting account holders.

In a statement Brigid Simmonds, chair of the Betting and Gaming Council, said: “Our members did not seek exclusivity for the rights to screen FA Cup games. They are therefore happy for IMG to offer the rights to screen these games to the Football Association or another appropriate body so that the games can be viewed for free by the public with immediate effect.”

However it is not in fact within IMG’s remit to offer the streaming rights, as it bought only the betting rights from the FA. Finding a solution remains complex because the prospect of the FA streaming matches for free on its website could cause it difficulties with the BBC and BT Sport, which have paid for the broadcasting rights.

The FA had said that from the 2021-22 season its new broadcasting deal, which will see the FA Cup broadcast on free-to air television by the BBC and ITV, not BT Sport, will also allow it to show matches not picked by the broadcasters on its own website. So the exclusivity of the deals done by the seven UK betting companies will no longer apply from then anyway.

MPs from all parties lined up on Thursday to condemn the FA’s commercial tie-up with the gambling companies after an urgent question tabled by the Labour MP Carolyn Harris, who branded the deal “shameful”. Several MPs expressed concern about gambling’s hold on football and criticised the FA for signing a deal that allowed betting companies to show FA Cup third-round matches, compared to just with two on free-to-air TV, to customers who opened an account or placed a bet.

Adams suggested that he was considering cutting government funding unless the FA found a way to pull the deals, responding to the call by the former Conservative leader Iain Duncan Smith for the government to do so by saying: “That will very much be on my agenda.”

Other government sources said, however, that was extremely unlikely because the money paid via Sport England, £14.6m from 2017-21 and £2m annually for coaching, supports grassroots football participation programmes and is not conditional on governing bodies’ commercial arrangements.

“I’ve been very clear that the government has asked the FA to look at all avenues to review this element of their broadcasting agreement,” said Adams, telling MPs that he had summoned the FA for a meeting next week and would “see what opportunities there are to rescind this part of the deal”.

Audience numbers for the third-round matches shown on gambling websites are not publicly available as only the betting companies themselves have the data, but they are thought to be relatively small, given the games available are those not picked by broadcasters for a wider audience, and not kicking off at 3pm on a Saturday. The FA had pointed out that this also means that from the fourth round on, there are very few matches available to betting companies, because almost all non-3pm Saturday matches are shown on television.

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The discussion in parliament saw widespread support from all parties for reform of the Gambling Act passed in 2005 under Tony Blair’s Labour government, which liberalised regulation of the betting industry and has led to the explosion in advertising by gambling companies, and proliferation of offers and marketing.

The Conservative election manifesto committed the government to a review of the act, which it called “analogue legislation in a digital age” and Labour has also called for the act to be replaced.

For those concerned about their gambling, or that of a loved one, there is free help and advice available online at BeGambleAware.org or by calling the National Gambling Helpline on 0808 8020 133.