Mr Straw told a Home Office conference that an all-inclusive team would have better chances for success in world tournaments than separate entrants.
But Downing Street dismissed his comments as "personal" views, saying the Prime Minister does not foresee any changes in the current system.
Mr Straw's suggestion is also likely to be unpopular with the four national football authorities, who have made it clear they intend to continue fielding international sides.
Mr Straw's call, to an organisation called Network - which provides support for ethnic minority staff in the Home Office - was said to be part of a drive to create a more "inclusive" sense of being British.
He is quoted in The Times as saying: "The one area of sport where we have been consistently successful is athletics and that is the one area of sport where we do not have an English, Scottish and Welsh team but a British team.
"I personally look forward to the day when we have a British football team. I think we might start winning some games."
A Home Office spokesman added: "The Home Secretary made it clear he was expressing a personal view.
"He was talking in the wider context of a question about Britishness/Englishness and how he felt Britishness was a more inclusive word to use."
Weakening
A spokesman for the FA said in response: "We have always competed as individual nations and I think the majority of the public is also very much in favour of this set up."
He also pointed out an all-Britain team would mean a weakening of British influence in world football as each national association has its own vote inside Uefa and Fifa, the ruling bodies in European and World football respectively.
And a spokesman for the Scottish FA added: "It's a complete non-starter with no grounds of support whatsoever in Scotland."