Mr O Cuiv's demand came as Mr Coveney refused to disclose the latest test result on burger samples until they have been further investigated by an Irish laboratory.
The Fianna Fail spokesman said there were "a lot of rumours going around" and the people needed an opportunity to question the minister.
He suggested that Mr Coveney or his officials should give a weekly briefing to a Dail Committee which would "put a lot of people's minds at ease".
The minister confirmed last night that he has received test results on 24 samples from Silvercrest Foods in Co Monaghan but says the samples are now being sent to an Irish laboratory for further confirmation.
In a statement, the minister said 130 samples had been taken from meat and other products in the factory and sent to a laboratory in Germany during the past week.
He described the investigation as "complex" and said he would not be drawing any conclusion until he got the full results, adding that further test results were expected over the weekend.
"I am not prepared to draw any conclusion until I'm fully satisfied that such conclusions are supported by facts," he said.
"In any food incident the emphasis is always on dealing with the matter thoroughly, promptly and as transparently as possible to ensure that consumer confidence in the integrity of Irish food production is maintained."
His decision came as British supermarket chain Waitrose became the latest retailer to withdraw beefburgers from its shelves because of the controversy.
The chain withdrew frozen burgers made by Dalepak -- one of the firms at the centre of the investigation. It said it was taking them off the shelves "as a precaution".
Tests had already shown that Silvercrest Foods and Dalepak Hambleton in Yorkshire supplied beefburgers with traces of equine DNA to supermarkets, including one product classed as 29pc horse.
On Wednesday fast-food chain Burger King said it had decided to replace Silvercrest Foods as its suppliers.
It said it was sourcing other approved suppliers to replace burgers produced by Silvercrest. It did not say whether this was a temporary arrangement.
By now 10 million burgers have been taken off supermarket shelves across Ireland and Britain.