The airline, whose main business is UK-to-U.S. flights and which is owned by Richard Branson's Virgin Group and U.S. airline Delta, has refused to negotiate with the Professional Pilots Union (PPU) in a deal over benefits, the union says.
The PPU represents 450 of the company's 800 pilots but Virgin Atlantic has only negotiated with rival union BALPA, says the PPU.
"(The) union is threatening strike action unless the company include them in the negotiations and scrap the (benefits) review," the PPU said in a statement on Thursday.
A spokeswoman for Virgin Atlantic said there was currently no strike action planned and added that it could work with the PPU in future if it dropped a demand.
"We've always been willing in principle to recognise the PPU, but we won't agree to their demand that we derecognise BALPA, and disenfranchise the hundreds of pilots who belong to this union," she said in an emailed statement.
Air France-KLM is due to buy 31 percent of the airline from Virgin Group, making it the second-largest shareholder after Delta, in a deal expected to complete in 2019.
(Reporting by Sarah Young; editing by Stephen Addison)