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.
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.
News of the crackdown was broken by the Russian daily Novaya Gazeta, which specialises in human rights investigations.
Edited by sierraleone, 13 April 2017 - 07:04 PM.