Action to stem BSE began in 1988 when a ban on cattle food made from animal remains, including meat and bone from cows, was introduced. But the government of the time stopped short of outlawing all animal by-products.
The food health watchdog also wants a mass screening programme to test whether BSE has spread from cattle to sheep.
FSA chairman Sir John Krebs told a press conference that whether or not sheep were contracting BSE was an unanswered question. "We simply do not know," he said.
"Of the 40 million sheep in Britain, some 4,000 do succumb annually to another disease, scrapie, which appears not to have any human health risk. It is possible, however, that some of these animals are actually suffering from BSE."
Recycled meat
The recommended measures are contained in a draft review of current BSE controls ordered by the government.
Action to stem BSE began in 1988 when a ban on cattle food made from animal remains, including meat and bone from cows, was introduced. But the government of the time stopped short of outlawing all animal by-products.
The FSA has now called for the current ban on feeding recycled meat and bone meal to livestock to be extended so that it includes blood, gelatine and tallow.
It may also prevent the feeding of feathers and chicken manure to poultry and even fish waste to fish, although this has yet to be discussed.
'Cannibalism'
The report says that "intra-species recycling" - feeding animals the remains of other animals of the same species - could theoretically amplify a new BSE-like disease in an animal.
The report also recommends that as a matter of "great urgency" there is a need to develop a rapid screening method to test sheep for BSE.
A spokesman for the National Farmers Union said farmers would have to face the "terrifying prospect" of entire flocks being destroyed if BSE was discovered in sheep.
"The NFU would be letting sheep farmers down, and farmers would be letting themselves down, if they did not contemplate this worst-case outcome," he said.
Sausage casings
Sheep have suffered from scrapie for about 200 years but there is no evidence that the disease is harmful to humans.
However, scientists have shown that, under laboratory conditions, BSE can infect sheep as well as cattle. And the FSA said new research should also be conducted to assess the possibility that cattle and sheep may carry BSE without showing symptoms.
Scientists demonstrated recently that a sub-clinical form of BSE could theoretically exist.
The report also calls for a study to establish conclusively whether BSE can be transmitted in milk, and tests on sheep intestines used for sausage casings.
The FSA pointed out that the problems of cross-contamination of animal feed were likely to occur in other countries. It urged the EU Commission to take action, especially in countries with a known risk of BSE.