Mark Drakeford said the virus at the Anglesey factory was down to its internal organisation, but it had been brought into the Wrexham factory.
2 Sisters said “the health, safety and well-being of our colleagues is ultimately the thing that matters most at our business…
“We will not tolerate any unnecessary risks – however small – for our existing loyal workforce at the facility.”
It said it had no agency workers at Llangefni.
Meanwhile Rowan Foods, which employs 1,500 people, said 38 staff were “absent due to testing positive for Covid-19”, but said there was “no clear evidence to suggest that there is a spread of the virus within the site”.
Mr Drakeford told BBC Radio 5 Live: “What we’re being told is that there’s a different explanation in Wrexham and a different explanation on the island of Anglesey.
“On Anglesey, we think it may be to do with the internal organisation of the factory and the challenges of maintaining social distance.
“In Wrexham, so far the science is saying that wasn’t the cause, it’s not internal to the factory. It’s just that the disease was brought in by people who would become infected outside the factory so there’s not just one explanation for both.”
He said ministers did not believe the Wrexham factory had found social distancing a challenge.
In Wales, employers are required by law to do everything they can to make sure workers can socially distance while in the workplace, or face a fine of up to £120.
The Food Standards Agency said it was “very unlikely you can catch coronavirus from food” as the virus is a respiratory illness.
2 Sisters is one of the largest food producers in the UK and produces about a third of all the poultry products eaten each day.
It has suspended production and closed the factory, which supplies local authorities, hospitals, restaurants and small businesses, following the outbreak.
The company also supplies food to KFC and supermarkets including Marks & Spencer, Aldi, Asda, Co-op, Morrisons, Sainsbury’s, Tesco and Waitrose, but not from the Anglesey site.
Public Health Wales said staff and contractors working at the processing plant, which has 560 workers, had been asked to self-isolate for 14 days.
It said staff would be tested, with all workers being contacted by the company to arrange the tests.
Rowan Foods, which is owned by Oscar Meyer Quality Foods, makes ready meals for companies including Aldi, Asda, and Sainsbury’s.
Anglesey MP Virginia Crosbie said she was “hugely concerned” by what had happened a 2 Sisters.
“This situation highlights the risks faced by people working in this sector and we should all be hugely grateful for the contribution that the 2 Sisters employees have made, and the courage that they have demonstrated, in supporting the fight against Covid-19,” she said.
She said she was liaising with the management at the plant to ensure that “none of the workers are left without pay during the duration of the plant’s closure”.
“People are scared at the moment and it is extremely important we are able to bring people with us in this discussion,” she added.
“We recognise the value of the tourism sector here on Ynys Môn, but also the people of Ynys Môn deserve to be protected as well.”
Anywhere cold, damp and indoors is an ideal environment for the coronavirus to thrive.
It survives best on cool surfaces, especially if there’s no dry breeze to get rid of the moisture or any ultraviolet light from the sun to kill it off.
Add to that the challenges of social distancing on a busy production line, together with loud machinery forcing staff to raise their voices.
Researchers know that situations where people sing – or have to shout – increases the chances of them projecting the virus to others nearby.
According to Prof Calum Semple, an infectious disease specialist at the University of Liverpool, and an adviser to the government, meat processing plants can be “a perfect environment for the virus to persist on surfaces and in the air.”