A government spokesman called the reports “lies” and said homosexual people “simply don’t exist in the republic”.
Homophobia is widespread in Chechnya.
Ms Poplevskaya told the BBC that victims – either gay or just perceived as gay – are being held at a detention centre near Argun, 20km (13 miles) from the city of Grozny.
She said the LGBT Network, based in St Petersburg, was aware of “an organised campaign to detain gay men” in Chechnya.
More than 30 people were crammed into one cell, she added.
Kadyrov spokesman Alvi Karimov denied the alleged persecution.
“You can’t detain and repress people who simply don’t exist in the republic,” he said.
The gay rights group has an email hotline to help victims, or potential victims, flee to other parts of Russia. Ms Poplevskaya declined to say where those victims were being sheltered.
News of the crackdown was broken by the Russian daily Novaya Gazeta, which specialises in human rights investigations. It reported that the detainees included some influential Muslim clerics close to Mr Kadyrov, and two well-known Chechen TV presenters.
“We confirm what was said in Novaya Gazeta,” Ms Poplevskaya said, while declining to name her sources in Chechnya.
Vigilantes claiming to be enforcing Muslim morals have long targeted gay people in Russia’s North Caucasus.
But persecution of gay people in Chechnya intensified in late February, she said.
LGBT Network submitted reports about the crackdown to the Russian prosecutor-general’s office, the Federal Investigative Committee (SK) and federal commissioner for human rights.
“We got no response, despite all the appeals. The only thing was that the Russian ombudswoman said she would initiate an investigation. That was only after Amnesty International filed their own statement,” she said.
But it is not a “concentration camp” for gays, she said, rejecting the description used in some media reports.
Foreign governments and human rights groups have voiced concern about the alleged abuses, urging the Russian and Chechen authorities to stop them.
The EU, the US state department, Human Rights Watch and Amnesty International are among those who have complained.
Reporting by the BBC’s Laurence Peter