Some teams also scan new players as part of a pre-signing medical.
"There is some feeling in football that players should be screened more," said Dr Ian Beasley, the Football Association's head of medical services.
But Beasley told BBC Sport that even if agreement is reached at a meeting of FA medical experts on 3 May and amongst Premier League club doctors on 10 May, the funding for an extended scheme would still have to be found.
"That is something we would have to work out how to fund," he added.
"Whether that is from the Premier League or from a national institution, that is something to be decided later on.
"There are funding issues in this. It is not just about football - it is about the whole of the athletic population.
"If it was decided by government and public health that screening in young people was beneficial and saved enough lives - don't forget in this country there is a cost-benefit analysis of health care - then the government should think about funding it."
Beasley, who previously worked for Arsenal for six years before joining the FA in December 2008, added that the screening process is not foolproof.
While the England team are tested as a requirement of their participation in Uefa and Fifa tournaments, Beasley said Muamba was given a clean bill of health by a heart screen in August.
"We do 800 tests per year and one to two are advised to give up football," he added. WHAT ARE THE TESTS?
Most cardiac abnormalities can be diagnosed by a cardiologist using an ECG (electrocardiogram). For extra clarity an Echocardiogram (a type of ultrasound scan) can also be undertaken.