The migrants were set to embark from Sabrata beach, 70 kilometres west of Tripoli, in a perilous attempt to cross the Mediterranean, the source said.
They were killed after refusing to board their boat due to bad weather, the source added.
The Libyan Red Crescent confirmed the killings had taken place over the weekend, but was unable to provide further details.
The Geneva-based International Organization for Migration said Tuesday it had heard reports of the deaths.
"There seems to have been an exchange of fire between the smugglers, which resulted in the death of 22 migrants," it said.
Smugglers organise frequent crossings from western Libya to the Italian island of Lampedusa, a 300-kilometre (190-mile) trip.
The IOM said 521 immigrants died in the Mediterranean from January 1 to March 5, 2017.
Another 20,000 were able to reach the European coasts during the same period.
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