Members of the public are to be asked if they find low levels - or "trace contamination" - of horse and other species in beef products acceptable.
The Food Standards Agency says that, while 20 beef products tested had more than 1% horse DNA, "very low levels" of horse and pig were found in others.
Chief executive Catherine Brown said consumer forums would be asked if it was "acceptable at certain low levels".
She said the government was exploring "implications for faith groups".
It was acknowledged that, for some religious groups, "any level of trace contamination, however low, is unacceptable", she added.
Last week, four beef products sold by Bird's Eye, Taco Bell and catering supplier Brakes became the latest found to be affected by "gross contamination" - with more than 1% horse DNA.
The discovery of 20 such contaminated beef products, in total, follows more than 6,000 tests in six weeks.
"In addition to the cases of gross contamination there have been cases of trace contamination," Ms Brown added.
She said trace contamination with other animals - including pig and horse - could take place at abattoirs and processing plants dealing with more than one species "even with thorough cleaning and good hygiene practice".
The FSA was using the 1% threshold "as a pragmatic level to determine the difference between gross and trace contamination", she said.
"The question that we want to explore is what levels are achievable, detectable and acceptable."
She said the Laboratory of the Government Chemist was looking at what levels of cross-contamination could take place in "a well-run and hygienic plant" and testing methods were also being looked at.
"Finally, there is a question of consumer acceptability - we need to have a better understanding of how consumers view trace contamination," she said.
"Is it acceptable at certain low levels? If not what are the trade-offs between costs and trace?"
The FSA was conducting a series of forums with consumers to try to answer those questions, she added.
The agency says figures are not currently available for cases of trace contamination of other species found in beef products.
Ms Brown said "about half" of the 1,527 adult consumers who completed an online survey carried out by the FSA said they would now purchase less processed meat or ready meals.
Of those, 67% gave a lack of trust as a reason while 35% said they did not want to eat horsemeat.
The survey was carried out between 15-18 February.
It is not clear if any cases of trace contamination referred to by Ms Brown related to halal products.
In February, pies and pasties distributed to prisons described as halal were found to contain traces of pig DNA.
Halal meat is defined as meat slaughtered by hand and blessed by the person doing the killing although some Muslims believe a mechanised form is also now acceptable.
Under Islamic law, the eating of pork is strictly prohibited for Muslims, as is the consumption of meat which has not been slaughtered in the correct way.
Muslim News editor Ahmed Versi said any incidents of trace contamination were, nevertheless, "a real concern" for the Muslim community.
"Many Muslims are not buying processed food now because they are concerned there might be contamination," he said.
"Even if it's horsemeat, it's not halal, and it might be pig which is their main concern."