Natalia Poplevskaya of the Russian LGBT Network said "we are working to evacuate people".
A government spokesman called the reports "lies" and said homosexual people "simply don't exist in the republic".
Homophobia is widespread in Chechnya.
The mainly Muslim region is run by Ramzan Kadyrov, an authoritarian leader with a notorious private militia, fiercely loyal to Russian President Vladimir Putin.
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.
"Torture is going on with electric shocks, beatings with cables. All the people arrested are homosexual men or perceived as being gay," she said. Three deaths have been reported.
More than 30 people were crammed into one cell, she added.
Despite appeals to the Russian authorities to stop the abuses, no action has been taken, she complained.
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.
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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.
"The office of the military commandant is now the unofficial detention centre for torture, near Argun. All the victims confirmed that," 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

Protesters hung a sign over the city's famous Rialto bridge and some carried suitcases to symbolise having to leave.
Locals say they are being forced out by high rents, housing shortages and a surge in the number of apartments being rented to visitors at inflated prices.
The population of Venice has dropped to under 55,000, from about 175,000 in 1951.
Some of the protesters were received by the city hall and later said they were encouraged by the meeting.
The number of people in the picturesque and car-free lagoon-side city swells in the daytime, especially in peak season when as many as 60,000 tourists can visit.
Earlier this year, people on small boats tried to stop a cruise ship from docking.
The city's mayor has proposed taxing day trippers to help deal with the problem.
"Venice is losing 1,000 residents each year," said Matteo Secchi, head of the association venessia.com which organised the rally.
"We are turning into Pompei, a town which people come to visit and say it's magnificent, but no-one lives there."
Gay men are fleeing brutal persecution in Chechnya, where police are holding more than 100 people and torturing some of them in an anti-gay crackdown, Russian activists say.
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Hundreds of residents of Venice have staged a protest against the rapid depopulation of the Italian city.