First it was Martina Navratilova. Then Paula Radcliffe and Kelly Holmes. Now, almost with every passing day, there are more voices wading into the issue of whether transgender competitors should be allowed to take part in women’s sport.

Over the weekend, the swimmer Sharron Davies, a silver medallist at the 1980 Olympics, became the latest to lend her support to a campaign led by the pressure group Fair Play for Women. “I believe there is a fundamental difference between the binary sex you are born with and the gender you may identify as,” Davies said. “To protect women’s sport, those with a male sex advantage should not be able to compete in women’s sport.”

Her remarks drew criticism from Rachel McKinnon, a transgender cyclist who won a Masters Track World Championship title in October, who said Davies was a “transphobe” and was “sharing hate speech”.

“There is no debate to be had over whether trans women athletes have an unfair advantage: it’s clear that they don’t,” McKinnon wrote. “Since the Nov 2003 IOC policy openly allowing trans women to compete, not a single trans athlete has even qualified for the Olympics, let alone won a medal.”

But Fair Play for Women insists women are clearly disadvantaged at both elite and grassroots level when they compete against trans athletes.

“Claims of transphobia are wild and unfounded,” its spokeswoman Dr Nicola Williams told the Guardian. “It is a tool used to shame women into silence. We want to get past not being able to speak out and to encourage a debate that leads to sports bodies implementing policies that are fair to women.

“We have actively tried to reach out to high-profile former athletes like Sharron and Martina to say: look, we have a problem here. Because I know of some women athletes who have been told they will have their sponsorship removed if they speak about this issue which could threaten their careers.”

All scientists agree that testosterone during growth, puberty and maturation results in large differences between the sexes. The debate is whether those advantages are lost if a trans gender athlete takes medication to reduce that testosterone.

As things stand, the International Olympic Committee allows trans women to compete as long as they have been reducing their testosterone levels for 12 months. However, Fair Play for Women believes there is little science behind the IOC’s policy and says there is a “legacy effect of testosterone” that gives an unfair strength advantage even after the current level of the hormone is reduced.

Governing bodies are clearly undecided about how best to deal with the questions of gender, biology and identity that arise from the issue. Last year Hannah Mouncey, a former member of the Australian men’s handball team, was blocked from playing in Aussie rules football’s professional women’s league on the grounds of strength and physique.

However, Laurel Hubbard, a New Zealand weightlifter who competed in men’s competition before transitioning at the age of 35, was allowed to take part in the 2018 Commonwealth Games.

The issue is also of increasing importance in schools and grassroots sport. Some US states, for instance, allow people to self-identify as women without reducing their testosterone, which Williams says could impact on others getting sporting scholarships for college, which would be unfair and discriminatory.

“Self-identification policies, where someone simply has to declare themselves as female, will have a impact on the women and girls competing now and in the future,” she said. “It is so vastly unfair. Why would you want to compete in a sport where you literally can never win?

“The problem is national governing bodies in the UK don’t have a policy on this. Whenever I speak to them they say the policy is under review – they are always waiting for someone else to make the first move. We are in limbo now and it is causing problems.”

On Sunday, Navratilova, who had a trans coach, apologised for saying a fortnight ago that trans athletes were cheating. “I’m sorry for that because I certainly was not suggesting that transgender athletes in general are cheats,” she said. “I attached the label to a notional case in which someone cynically changes gender, perhaps temporarily, to gain a competitive advantage. We should not be blind to the possibility and some of these rules are making that possible and legal.”

Both Navratilova and Davies are calling for sports bodies to debate the issue. However, the LGBT rights group Stonewall insists the sports bodies must be fair on transgender athletes, too. “Sport should be welcoming to everyone, including trans people,” it said. “We need clubs and governing bodies, as the experts, to consider how their sports’ individual policies can work to be as inclusive as possible, to ensure all people, including trans people, can take part in sport.”

Radcliffe, meanwhile, who has previously said trans women being able to compete “makes a mockery of the definitions of male and female sports categories”, said: “People are scared to talk about the issue. But I believe protecting women’s sport is important.”