Earlier this month, British Prime Minister David Cameron invited England's women's soccer team to Downing Street to congratulate them on their roaring success at the World Cup.

Mingling in the premier's leafy back garden, several of the players, with their bronze medals twinkling in the sunshine, told Cameron they hoped they would be playing in next summer's Olympic Games in Rio de Janeiro.

Those Olympic dreams had gotten a big boost in Canada after the squad returned home to their soccer-crazy nation with a medal — a rare achievement for England in international competitions.

The Lionesses, as they are known, lost a semifinal match against Japan after scoring a gut-wrenching goal in their own net, but their overall performance and third-place finish should have been enough to secure Great Britain a spot in the Rio Games.

And yet, it now seems that political squabbling among the four nations that make up the United Kingdom — England, Wales, Scotland and Northern Ireland — will likely snuff out the women's hopes.

At the heart of the debate over whether Britain will field any soccer teams at the Olympics are questions about British identity, and which of Britons' multiple identities gets priority.

The four constituent nations of the United Kingdom compete as individual teams in tournaments such as the World Cup and the European Championships. But in the Olympics, the athletes must compete under the single banner of "Team GB."

FIFA, the world governing body of soccer, said that Britain would need to submit a bid for the Olympics with the support of all four of the national soccer associations, but Scotland, Wales and Northern Ireland are against the idea. They think it would damage their prospects of retaining nation status within FIFA, and their ability to compete as individual nations in other international tournaments.

"We want to maintain our position as an independent football nation, and we believe that playing in a tournament as Team GB could undermine our position long term," Ian Gwyn Hughes, a spokesman for the Football Association of Wales, said in an e-mail.

The wrangling comes amid growing waves of nationalism within the union and especially in Scotland, which came within 10 percentage points of leaving the United Kingdom in a referendum last year. Since then, the Scottish National Party — whose main goal is the break-up of the union — has become much more powerful, winning 56 out of 59 seats in Scotland during the last general election.

"To start appearing on a team that's called 'Great Britain' I suspect would seem for many nationalists in Scotland to be an admission of something they are trying hard not to admit," said Tony Travers, a political expert at the London School of Economics.

The sporting rivalry among Britain's nations can be intense, particularly between England and Scotland, where the neighborly antagonism predates the upsurge in nationalism. Andy Murray, the Scottish-born tennis player, once famously said he was cheering for "anyone but England" in an international competition — a common sentiment in Scotland.

Earlier this year, the Football Association, the soccer organizing body in England, proposed sending a team to the Olympics, but after failing to get backing from the other nations, the idea was binned in March.

For some players, it was devastating news. "Being Welsh and maybe having an opportunity to represent Great Britain at an Olympics would've been a dream. Now that's taken away," tweeted Welsh player Natasha Harding.

The landscape was different in 2012, when Team GB fielded men's and women's soccer teams at the Olympics in London — the first time since the 1960 Games in Rome.

To some soccer fans, the decision not to come together for Rio smacks of sourness by the non-English nations whose soccer teams have a lesser track record at international tournaments.

But others say that there is a sense of arrogance on the side of the English, and that it was widely assumed that the decision to field a team in London was a one-time occurrence because the competition was on home turf.

"There was never any plan to have a permanent British team and we made that very clear, as did the Welsh and Northern Irish," Stewart Regan, chief executive of the Scottish Football Association, told the BBC.

England has claims to inventing the game of soccer, and on the men's side, it is wildly popular, with England's Premier League one of the most popular in the world. The Olympics, which on the men's side has an age restriction of younger than 23 (with the exception of three players), is arguably not as important for the men as other tournaments. And the men would not have qualified for Rio.

But the sport is still developing for the women, and some fans say it's disappointing that the women won't get the sort of mega-watt exposure a platform like the Olympics can offer.

"Everyone I know is gutted," said Sophie Downie, a blogger for Girls on the Ball, a Web site that follows women's soccer in the United Kingdom.

Colin Moynihan, the former chairman of the British Olympic Association, recently took up the issue in the House of Lords, where he is a member.

He has said that four years ago he received assurances from FIFA that the status of the home nations would not be under threat if they entered a Team GB in the London Games.

When asked whether it had given similar assurances over the Rio Games, FIFA said in an e-mail: "We have no further comment."

"The home nations want to hold on to their privilege and status within FIFA, and there are those who want to challenge it on the grounds: Is Great Britain one country or four nations? There are tensions where this comes up," said Grant Jarvie, an expert on sport at the University of Edinburgh.

As it stands, it seems unlikely that the women will be joining other British Olympians in Rio. "We feel the boat has sailed on this one," said Miriam Wilkens, a spokeswoman for the British Olympic Association, noting that various deadlines have passed. But she said they were hopeful that something could be worked out for the 2020 Games in Tokyo.

When it was explained to Cameron at the garden reception that it wasn't clear that any of the women he was congratulating would be competing at the next Summer Games, according to British news agency Press Association, he said: "That's frustrating. Disagreement between the home nations? Can we fix this?"