The roundup has swept up men from across the republic, including two well-known local television personalities, the newspaper, Novaya Gazeta, reported.
The men disappeared "in connection with their nontraditional orientation or suspicion of such," the paper said. It said that it has learned the names of at least three men who were allegedly killed and that it suspects there have been many others but did not provide further details.
The government has denied the allegations.
Ekaterina L. Sokiryanskaya, the Russia project coordinator for the International Crisis Group and another expert on the North Caucasus, told The New York Times her sources alerted her to the alleged purge.
A majority-Muslim republic devastated by two separatist wars with Russia since 1994, Chechnya is regularly the scene of alleged brutal human rights abuses, with its security services accused by rights groups of carrying out extrajudicial killings and torturing opponents of its president, Ramzan Kadyrov.
It was already a dangerous place to be gay, where attitudes on homosexuality are deeply conservative and gay men are often shunned as bringing dishonor to their families, with anti-gay attacks common. Kadyrov's government has recently launched a drive promoting what it calls traditional values, including alleged castigating of gay men.
Alvi Karimov, a spokesman for Kadyrov, on Sunday rejected the Novaya Gazeta story as "absolute lies," saying it was impossible because Chechnya has no homosexuals.
"You cannot arrest and repress those who simply aren't in the republic," he told the Russian news agency Interfax. "If there were such people in Chechnya, then the law enforcement agencies wouldn't have to worry about them, as their own relatives would have sent them to where they won't return."
According to the paper, the group did not apply in Chechnya but in a neighboring republic, Kabardino-Balkaria. The application, as usual, was refused but was nevertheless met by a wave of anti-gay counterprotests.
"Precisely at this time, the command was given for the 'prophylactic purge,' and it went as far as real murders," Novaya Gazeta wrote, citing sources in the Chechen security services.
Novaya Gazeta and a rights group, the Russian LGBT Network, said they were submitting a formal request for Russia's federal authorities to open a criminal investigation into the alleged detentions and killings.
There is skepticism whether federal authorities will intervene, given that Russia has passed legislation discriminating against LGBT people. The so-called gay propaganda law prohibits the promotion of "nontraditional" sexual orientations to minors, punishable by a fine or short prison sentence.
Kremlin spokesman Dmitry Peskov Monday said that officials were unsure "to what extent the information was true" and that it was not the Kremlin's role to investigate.
"It is not a prerogative of the Kremlin," he said, according to Interfax. "If any actions have been taken by the law enforcement agencies which, in the opinion of some citizens, were taken with some irregularities, these citizens can use their rights, file relevant complaints and go to court."
Confirming details of the disappearances is difficult in the region, where speaking out against authorities can result in retaliation. But Igor Kochetkov, the director of the Russian LGBT Network, told ABC News that his network has received dozens of requests for help since February — which appears to tally with the campaign reported by Novaya Gazeta.
"We have information that there are dead," he said, adding that his group is looking further into the stories. "You need to understand that people are very frightened."
Kochetkov's network and another rights group have established hotlines for people feeling threatened and are offering aid to help get people out of Chechnya.
Lokshina said the Russian government should intervene. "What's been happening is absolutely lawless," she said.