Lizzie Wilson, policy services officer at the National Pig Association (NPA), said the shortage of butchers meant processors were operating at 25% reduced capacity.
As a result mature pigs ready for processing are backing-up on farms, causing welfare issues.
"There's about 120,000 pigs sat on farm currently that should have already been slaughtered, butchered, be within the food chain and eaten by now," said Wilson.
"It is getting to the point where we are saying to government if we don't get some help soon we're going to have to look at culling pigs on farm, because that's our only option now," she said, adding "there are some producers that have already had the conversation."
Britain's biggest pork processors are Cranswick, Morrisons, Pilgrim's Pride and Karro Food Group.
Wilson said consumers were already seeing the impact of the crisis on supermarket shelves as processors had rationalised pork product ranges.
Minette Batters, president of the National Farmers Union, said a cull of up to 150,000 pigs was "potentially a week, ten days away".
"I do not feel anybody can preside over a welfare cull of healthy livestock. I don't believe it has happened in the world before and it cannot happen now," she told the BBC.
Batters said she wants an urgent meeting with interior minister Priti Patel and immigration minister Kevin Foster.
She said she has been trying to get a meeting with Patel for two years.
David Lindars, technical operations director at the British Meat Processors Association, said a cull "was getting very close."
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