Up to 100 men are reported to be held in the camps, with at least three having died there.
Alvi Karimov, a spokesman for Chechnya's leader Ramzan Kadyrov, dismissed the claims as "absolute lies and disinformation", The Huffington Post noted.
The Russian republic of Chechnya has reportedly opened the first detention camp for homosexuals since the Nazi-era.
One of those who escaped told Novaya Gazeta that prisoners were beaten to force them to reveal other members of the gay community.
"The detention and ill-treatment of over 100 gay men in Chechnya is extremely concerning", she said. We are also outraged by the reaction of the officials of the Chechen Republic, who in fact justify the killings.
We call on the authorities immediately to launch an investigation that is effective and fully in line with all the requirements of the European Convention on Human Rights.
His spokesman Alvi Karimov told Interfax news agency "you can not arrest or repress people who just don't exist in the republic".
"Poplevskaia said the Russian LGBT Network has begun helping gay and bisexual men flee Chechnya but did not want to describe details of the evacuation program out of safety concerns".
Chechen's President Ramzan Kadyrov, who is a key ally of Vladimir Putin, accused of ordering the clampdown, denied the allegations claiming "it's impossible to persecute those who are not in the republic".
One of Chechnya's human rights officials, Kheda Saratova, echoed those sentiments as he dismissed the report.
They report that trustworthy sources and witnesses have said three men have already been killed and perhaps a fourth. He even added that "if there were people like this in Chechnya, the authorities would not have to do this because their families would send them to a place from which they would not return".
The Russian LGBT Network makes every effort to contact the victims and to provide the emergency support.
While the Russian government's office for human rights has not confirmed the report, LGBT rights organizations in Russia have condemned it, and are trying to evacuate gay citizens at a risk of persecution.