The former Wimbledon champion Martina Navratilova has been criticised for “disturbing, upsetting, and deeply transphobic” comments after she argued that allowing transgender women to compete in women’s sporting tournaments was “insane and cheating”.

The tennis player and gay rights campaigner first drew criticism from equalities activists and trans athletes when she tweeted in December: “You can’t just proclaim yourself a female and be able to compete against women. There must be some standards, and having a penis and competing as a woman would not fit that standard.”

Writing in the Sunday Times, Navratilova said she had subsequently promised to keep quiet on the subject until she had done some research on it. “Well, I’ve now done that and, if anything, my views have strengthened,” she wrote.

“To put the argument at its most basic: a man can decide to be female, take hormones if required by whatever sporting organisation is concerned, win everything in sight and perhaps earn a small fortune, and then reverse his decision and go back to making babies if he so desires.

“It’s insane and it’s cheating. I am happy to address a transgender woman in whatever form she prefers, but I would not be happy to compete against her. It would not be fair.”

Her comments attracted criticism across social media. “We’re pretty devastated to discover that Martina Navratilova is transphobic,” tweeted the rights group Trans Actual. “If trans women had an advantage in sport, why aren’t trans women winning gold medals left, right and centre?”

Under guidelines introduced by the International Olympic Committee in 2016, trans men are able to compete without restriction, while trans women must demonstrate that their testosterone level has been below a certain cutoff point for at least one year before their first competition.

Previous guidelines, approved in 2003, required transgender athletes to have reassignment surgery followed by at least two years of hormone therapy in order to be eligible to compete.

“Simply reducing hormone levels – the prescription most sports have adopted – does not solve the problem,” wrote Navratilova. “A man builds up muscle and bone density, as well as a greater number of oxygen-carrying red blood cells, from childhood.”

Following her comments in December, Navratilova was criticised by Rachel McKinnon, a Canadian academic and cyclist, who in October became the first transgender woman to win a track world title.

“McKinnon has vigorously defended her right to compete, pointing out that, when tested, her levels of testosterone, the male hormone, were well within the limits set by world cycling’s governing body,” wrote Navratilova on Sunday. “Nevertheless, at 6ft tall and weighing more than 14 stone, she appeared to have a substantial advantage in muscle mass over her rivals.”

The tennis star said she had been “pretty put out” by McKinnon’s accusation that she was transphobic and said she deplored “what seems to be a growing tendency among transgender activists to denounce anyone who argues against them”.

She pointed to her friendship with Renée Richards, the transgender tennis player who campaigned to be able to compete in the women’s US Open, and her support for Caster Semenya, who is fighting a legal battle to be able to compete without taking testosterone-suppressing medication.

In a statement to the Guardian, McKinnon described Navratilova’s article as “disturbing, upsetting, and deeply transphobic”. “She trades on age-old stereotypes and stigma against trans women, treating us as men just pretending to be real women. She seeks to deny trans women equal rights to compete under the rules,” she said.

“And the current rules, such as the International Olympic Committee since 2003, explicitly welcomes trans women to compete at the highest levels. I suppose it’s too much to ask for Martina to simply do the same.”

Following McKinnon’s win in the women’s 35-44 sprint during the UCI Masters Track Cycling World Championships in Los Angeles in October, Jennifer Wagner, who came third in the race, complained that the result was unfair.

McKinnon responded by saying Wagner had beaten her in 10 of their last 12 races. “This is what the double bind for trans women looks like: when we win, it’s because we’re transgender and it’s unfair,” she said. “When we lose, no one notices (and it’s because we’re just not that good anyway). Even when it’s the same racer. That’s what transphobia looks like.”

A spokesperson for the LGBT rights charity Stonewall said: “Sport should be welcoming to everyone, including trans people. We need clubs and governing bodies, as the experts, to consider how their sports’ individual policies can work to be as inclusive as possible, and what advice and guidance they’re giving to ensure all people, including trans people, can take part in sport.”