The taxpayer-funded school for up to 60 pupils will be based in Manchester, if preliminary studies find the idea would work.
Supporters say the school will provide students protection from homophobic bullying, and its effects.
Amelia Lee, Strategic Director for LGBT Youth North West, said: "This is about saving lives.
"Despite the laws that claim to protect gay people from homophobic bullying the truth is that in schools especially, bullying is still incredibly common and causes young people to feel isolated and alienated, which often leads to truanting and, in the worst-case scenarios, to suicide.
"Unfortunately, schools can be one of the last bastions of homophobia.
"This would be somewhere that students who are struggling with the negative effects of issues like bullying could attend classes for a period of time while ensuring they get the grades they are capable of."
However the Conservative MP and former education minister Tim Loughton said: "We need to do a lot more to combat homophobic bullying and to create a more tolerant society.
"But I cannot see how segregating a group of young people identified by their sexuality can aid better engagement and understanding.
"The way to achieve more integration, understanding and empathy is not by segregating members of one group, and this would seem to me to be a step backwards from achieving tolerance."
According to the Daily Mail newspaper, a source "close" to Education Secretary Nicky Morgan said: "There is simply no way that we will approve a free school specifically for LGBT young people.
"Pupils regardless of their sexuality should be educated in mainstream schools which should be equipped to tackle any bullying that should occur."
The Church of England has previously expressed a desire to tackle homophobic bullying in its schools. Last year Archbishop of Canterbury Justin Welby launched a campaign to raise awareness in church schools.