A shortage of skilled butchers has left abattoirs unable to process any more pigs.
With farms struggling for space to hold onto the animals and the money to keep feeding them, many face having to destroy the pigs they have reared.
Steve Thompson, owner of Moss Valley Fine Meats in North East Derbyshire, said the impending slaughter and lack of support was having an emotional impact on pig farmers.
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On Monday he joined other pig farmers to demonstrate in Manchester close to the Conservative Party conference, to raise awareness about the crisis affecting their sector.
Wearing pig costumes and pig masks they held up placards which read 'We farm to feed the nation not feed the landfill' and 'We'd rather give it away than throw it away'.
Mr Thompson said: “The mental anguish of farmers is unbelievable at the moment – we’re a week away from having no room for pigs, from not being able to cope.
“When Boris did his interview with Andrew Marr, I nearly cried, and it takes a lot to make a farmer cry. It was so disheartening for him not to have a clue what he was being asked, to have no regard for animal welfare or the farming industry.”
As well as the distress the crisis is causing for farmers, Mr Thompson said it was creating financial difficulties for those involved in the industry.
“We’re losing a lot of money feeding pigs that are getting bigger and heavier, and then processors fine us for sending them overweight pigs, even though they didn't take them off us when they were ready - we can’t win. I don’t know where we go with it.”
In September, the National Pig Association reported an extra 85,000 pigs were being held on UK farms, a number that was set to rise by 15,000 each week. This means that nationally, as many as 145,000 pigs could be set for slaughter.
Andrew Critchlow, County Adviser for the Derbyshire National Farmers’ Union branch, said the issue was just one example of the worker shortages that have hit the UK’s agricultural sector.
“It’s symptomatic of the general problems facing agriculture at the moment, in terms of processing products. That’s where the issues are in terms of the labour, and we’re also seeing dairy farms struggling to get labour too.”
Mr Thompson urged the government to bring in temporary visas for butchers, to avoid sending thousands of pigs to landfill and to make sure customers can get pork at Christmas.
“There has to be temporary visas because they’ve let the problem get so big, this is the only way to get around it – or there’ll be no pork on the shelves at Christmas.
“The government has this idealistic idea of closing the borders, but there aren’t the people here to do the jobs that we need them to do.”
Despite recent reporting on the issue, Boris Johnson appeared unaware of the problem when questioned on The Andrew Marr Show, telling the presenter: “I hate to break it to you but I am afraid our food processing industry does involve the killing of a lot of animals. I think your viewers need to understand that.”