Betting companies with streaming rights for FA Cup ties say they would allow the games to be streamed on a free platform elsewhere.
The Football Association has been criticised for its decision to sell FA Cup broadcast rights via a third party to seven gambling websites.
Since the start of last season the bookmakers have been able to show FA Cup ties on their websites and apps.
BBC Sport understands the FA would be open exploring the possibilities.
It is understood the FA would not want matches shown to clash with other television broadcasts of live matches. There was a match broadcast in each of the kick-off slots during the FA Cup third round last weekend, except for the 15:01 GMT start time.
The seven gambling websites - Bet365, Betfair, William Hill, Coral, Ladbrokes, Unibet and Paddy Power - acquired the rights via agency IMG, who agreed a deal with the FA.
In the third round of the tournament, 23 matches were available to watch on Bet365 last weekend - all those that did not kick off at 15:01 GMT on Saturday.
The matches were available to anyone who has placed a bet or put a deposit in their account in the 24 hours before kick-off.
In July 2017, the FA announced it was cutting its ties with gambling firms, but the deal with IMG was made in January 2017.
Last weekend, ties started one minute late as part of the 'Heads Up' mental health awareness campaign.
The government is "very angry" about the issue and the sports minister Nigel Adams has said he will meet the FA next week.
But, Brigid Simmonds, chairman 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."
The FA has said it will "review this element of the media rights sales process ahead of tendering rights from the 2024-25 season", but the government want it to look at taking action earlier.