A spokesman for the British Olympic Association has said his organisation believes a boycott of the Winter Olympics in Sochi would serve only to "penalise athletes".
Darryl Seibel told ITV News that Olympic bosses are monitoring the situation in Russia closely and that the safety of athletes is a top priority.
A petition calling on the International Olympic Committee (IOC) to relocate the 2014 Winter Games away from Russia has garnered almost 130,000 signatures.
Petitioners want the Games to be taken away from Sochi because of laws in Russia which they say discriminate against LGBT people. They suggest relocating to Vancouver in Canada, which hosted the Games in 2010.
A Government spokeswoman said:
We remain greatly concerned about the growing restrictions on LGBT freedoms in Russia and have repeatedly raised our concerns, including at the 2013 UK-Russia Human Rights dialogue in May.
The Prime Minister outlined our concerns with President Putin during a meeting in Downing Street in June ahead of the G8 Summit.
We are working closely with the IOC and the BOA to ensure that the Games take place in the spirit of the Olympic Charter and are free from discrimination.
Stephen Fry has urged the Prime Minister to support a campaign to strip Russia of the 2014 Winter Olympics amid concerns about anti-gay laws passed in the country.
In an open letter on his website, the broadcaster said President Vladimir Putin "is making scapegoats of gay people" and "cannot be seen to have the approval of the civilised world."
Politicians in Russia's lower house of parliament, the Duma, have passed a law imposing heavy fines for providing information about homosexuality to people under 18.
Last month the IOC said it had "received assurances from the highest level of government in Russia that the legislation will not affect those attending or taking part in the Games."
Stephen Fry has thanked the Queen for giving royal assent to the Bill that passes same-sex marriage into law:
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Paul Farmer, the chief executive of mental health charity MIND, says Stephen Fry is helping to "sweep away the stigma surrounding mental health".
He said: "Stephen Fry, our president, has been extremely open and honest about his own mental health and the depths of despair he has experienced, and we know just from the response we've had in the last 24 hours how important that is to thousands of people who now feel that they're not alone."
There are around 6,000 people who take their own lives in this country every year.
For help and advice about mental health issues visit the MIND website.
Stephen Fry has told of the moment he tried to commit suicide, saying that "it was a close run thing."
The comedian and actor said he was only saved because his producer found him unconscious in his hotel room in 2012.
The Samaritans charity is available for anyone in any type of distress on 08457 90 90 90 in the UK or visit their website.
The actor and comedian Stephen Fry has shocked fans by admitting that he tried to commit suicide last year while filming overseas.
Fry, who suffers from manic depression which causes his mood to swing between euphoria and deep depression, said he was only saved because his producer found him unconscious in a hotel room.
Fry said he decided to speak out because he believed his role as president of mental health charity Mind compelled him "not to be shy and [to be] forthcoming about ... the likelihood of death amongst people [with] certain mood disorders."
Speaking of manic depression, he said: "I have a condition which requires me to take medication so that I don't get either too hyper or too depressed to the point of suicide."
Samaritans is available for anyone in any type of distress on 08457 90 90 90 in the UK or visit their website.