The warning comes as The Independent newspaper claims up to one in 30 horses being exported to Europe for consumption could contain traces of a drug, known as Bute, which is harmful to humans.
Environment Secretary Owen Paterson said the next set of results on all retailers’ and manufacturers’ processed beef products could reveal further traces of horse meat.
“There may well be more bad results coming through, that’s the point of doing this random analysis,” Mr Paterson said.
The results, ordered by the Food Standards Agency (FSA), are due on Friday.
But David Clarke, chief executive of Red Tractor Assurance, a food guarantee scheme that covers British production standards, urged people to put the scandal into perspective.
Paterson: ‘Prepare for more bad news’
He told Sky News: “The news in the last three weeks has been of great concern to consumers. But to get it in perspective it is only affected a small part of the food that is in the shops.
“I would hope that all of the fresh meat that people are eating for Sunday lunch today should not be affected by this.”
Mr Clarke added that the food industry had learnt lessons from the last few weeks, namely that “this very cheap processed meat produced with raw materials that are traded all across the world, all across Europe, is potentially a problem”.
Meanwhile, one of the food companies at the centre of the horsemeat scandal has said it is considering taking legal action against its suppliers.
Frozen foods firm Findus, which has taken its beef lasagnes off shelves after some were found to have up to 100% horse meat in them, said it was looking into legal action as an internal investigation “strongly suggests” that the contamination “was not accidental”.
The Ministry of Agriculture in Romania – to where the horsemeat has been traced – has launched an inquiry after two of its abattoirs were implicated in the scandal.
The Environment Secretary also revealed retailers have agreed plans to improved their food testing, adding that they hold the “ultimate responsibility” for making sure their products do not contain horse meat.
Mr Paterson was speaking after attending an emergency meeting with bosses from leading supermarkets, trade bodies and the FSA on Saturday to discuss the scandal which has seen chains including Tesco, Lidl, Aldi and Iceland withdraw some products.
He said supermarkets and trade bodies have already begun plans to carry out more testing and report their results on a quarterly basis.
Aldi has withdrawn products containing horsemeat
They had also agreed that consumers should be compensated if they have bought withdrawn products with no questions asked, he said.
Mr Paterson added: “It’s a question of either gross incompetence, but as I’ve said publicly and I’ll repeat again, I’m more concerned there’s actually an international criminal conspiracy here, and we’ve really got to get to the bottom of it.”
Scotland Yard have met representatives from the FSA, although there is currently no official police investigation.
Prime Minister David Cameron has described the scandal as “shocking” and “completely unacceptable”, while Labour leader Ed Miliband said it was “appalling”.
The Trading Standards Institute has said the discovery of such high levels of horse meat suggests “deliberate fraudulent activity”.
Food safety experts have said there is no risk to public health.
Tesco and Aldi have also withdrawn a range of ready meals produced by Comigel over fears that they contained contaminated meat.
The GMB union said all hospitals, schools and meals-on-wheels services should verify that horse meat had not been served to vulnerable people.
Responding to fears that school dinners might be contaminated with horsemeat, the Department for Education said schools and councils were responsible for their food contracts.