Hundreds of refugees and asylum seekers fled to Scotland and the rest of the UK from countries like Nigeria and Uganda due to persecution on account of their sexuality and gender identity.
The call from the Glasgow-based support group for those fleeing harassment due to bans on homosexuality follows demands yesterday for Scottish ministers to raise human rights abuses with their Commonwealth counterparts in the run-up to the Games.
An LGBT Unity Scotland spokesman said: "Our group includes people from Commonwealth countries such as Uganda, Nigeria, Namibia and Cameroon. who have experienced horrendous discrimination and persecution just because of their sexuality.
"Some of these men and women have fled for their lives leaving behind their homes, families, jobs, everything that was familiar to them, due to their governments' repressive and violent sanctions against same sex relationships.
"It's important these human rights abuses are not brushed aside when the Commonwealth Games come to Glasgow."
LGBT group Pride Glasgow called on Mohammad Sarwar, the former city MP and the new Governor of the Pakistan Punjab region, to use his influence to reform legislation in the country that bans homosexuality.
Pakistan has a potential life sentence for those in same-sex relationships. In 2003 in Lahore, the capital of the Punjab which is twinned with Glasgow, three men were arrested when one of their relatives said they had engaged in same-sex sexual acts. Their punishment is not known.
A Pride Glasgow spokesman said: "As with any country that persecutes LGBT people we would urge Pakistan to ensure the safety and wellbeing of its LGBT citizens. We would hope Mr Sarwar would use his influence to help ensure LGBT people in Pakistan are able to live free from discrimination, persecution or prosecution."
Last night the office of Prime Minister David Cameron, who previously cut aid to Malawi over its attitude to gay rights, also said the protection and promotion of the rights of lesbian, gay, bisexual and transgender people remained an integral part of the Government's wider international human rights agenda.
Several of the 41 Commonwealth countries which criminalise homosexuality carry the potential punishment of life imprisonment, such as Bangladesh, Sierra Leone, Tanzania, Uganda and Guyana.
Parts of Nigeria have the death sentence, while Uganda is considering legislation to introduce the death penalty for gay offenders.
Adult same-sex relations carry a potential sentence of 25 years in Trinidad and Tobago and 20 years and flogging in Malaysia. Since 1994 Malaysia has also banned gay, bisexual or transgender individuals from the state-run media.
A Downing Street spokesman said: "We can better challenge prejudice by being inclusive and using opportunities which bring people together to celebrate and promote our values of equality and diversity. The Commonwealth Games are first and foremost a great opportunity to showcase Scotland to the world and to build a legacy in terms of sport, participation and investment."
Only 13 Commonwealth countries do not criminalise same-sex relationships, while only five legally recognise them.