Activists are in talks with five countries, two of them non-EU, Russia's LGBT Network organisation told the BBC.
Nine men have already been granted visas. Two of them went to Lithuania, which has announced its involvement.
"It's very important to act, because they are suffering," Foreign Minister Linas Linkevicius told the BBC.
He would not name the other countries involved but described them as "allies".
His country's decision was an "implicit message" to Russia, he said, because "we are taking care of Russian citizens... [whose] rights were abused".
Reports of a brutal crackdown on gay men in the republic in southern Russia first emerged in early April through the respected Novaya Gazeta newspaper, sparking international condemnation.
Its report, later backed up by activists, alleged that more than 100 victims - either gay or just perceived as being gay - were being held and tortured at a detention centre near Argun, 20km (13 miles) from the city of Grozny. At least three deaths were reported.
The government of the mainly Muslim region has denied the crackdown and said homosexual people "simply don't exist" there.
Russian President Vladimir Putin has said he will ask authorities to look into the allegations.
The LGBT Network has helped to evacuate 43 people from Chechnya, with dozens remaining in hiding in Russia as negotiations continue with the countries that might take them in.
Neither Lithuania's foreign minister nor the LGBT Network would reveal the other countries involved. There have been calls in countries as far away as Australia for the men to be offered refuge.
An LGBT Network spokesperson said the UK, despite condemning the persecution, had not come forward to help with visas, while US embassy officials had said visa applications would probably be denied.
"We can't really lose time knowing that they are going to be denied," the spokesperson, who remained anonymous out of safety concerns, told the BBC. "They can't stay in Russia for too long, we know it's not safe for them to stay here."
German Chancellor Angela Merkel raised the issue on a recent visit to Russia but it is unclear if Germany is one of the countries involved.
The BBC has approached the UK Home Office for comment. The US State Department has said that Chechens can apply for visas but told Buzzfeed Newsthat there was no visa category specifically for "humanitarian relief."
The men would have to cross an international border to apply for refugee status.
Chechnya is run by Ramzan Kadyrov, an authoritarian leader with a notorious private militia, fiercely loyal to Russian President Vladimir Putin.
Earlier this month, three French gay rights groups filed a complaint at the International Criminal Court accusing Chechnya of a policy of genocide towards gay people.
They cited the case of a teenaged male thrown out of a ninth-storey window, allegedly because of his sexuality.