It follows a study examining the authenticity of a number of beef burger, beef meal and salami products available from retail outlets in Ireland.
The study, which tested for the presence of horse and pig DNA, revealed the presence of horse DNA in some beef burger products.
The FSAI said although there was no risk to public health, it raised concerns about the traceability of meat ingredients and products entering the food chain.
A total of 27 beef burger products were analysed, with 10 (37%) testing positive for horse DNA and 23 (85%) testing positive for pig DNA.
Some 31 beef meal products including cottage pie, beef curry pie and lasagne were also analysed. Twenty-one were found to have pig DNA while all were negative for horse DNA.
All 19 salami products analysed tested negative for horse DNA, but traces were detected in batches of raw ingredients including some imported from The Netherlands and Spain.
The beef burger products which tested positive for horse DNA were produced by two processing plants in Ireland, Liffey Meats and Silvercrest Foods, and one in the UK, Dalepak Hambleton.
However, in one sample from Tesco the level indicated that horsemeat accounted for approximately 29% of the content.
The FSAI says it is working with the Department of Agriculture, Food and the Marine, and the processing plants and retailers involved.
It said the retailers had pledged to remove all implicated batches from their shelves immediately. In addition, Silvercrest Foods was withdrawing all products from sale and replacing them with new ones.
In a statement, Tesco said the products found to have horse DNA included two types of frozen beef burgers sold by the chain in the UK as well as Ireland.
It said it "immediately withdrew from sale all products from the supplier in question".
It was working with the authorities in Ireland and the UK and the supplier to "urgently understand how this has happened and how to ensure it does not happen again."
It added: "The safety and quality of our food is of the highest importance to Tesco. We will not tolerate any compromise in the quality of the food we sell."
Professor Alan Reilly, chief executive of the FSAI, said although consumers need not worry, the findings did raise a number of concerns.
He said: "The products we have identified as containing horse DNA and/or pig DNA do not pose any food safety risk and consumers should not be worried. Consumers who have purchased any of the implicated products can return them to their retailer.
"Whilst, there is a plausible explanation for the presence of pig DNA in these products due to the fact that meat from different animals is processed in the same meat plants, there is no clear explanation at this time for the presence of horse DNA in products emanating from meat plants that do not use horse meat in their production process.
"In Ireland, it is not in our culture to eat horse meat and therefore, we do not expect to find it in a burger. Likewise, for some religious groups or people who abstain from eating pig meat, the presence of traces of pig DNA is unacceptable."
:: The FSAI operates an advice line on 1890 33 66 77 from 9.00am-5.00pm.