He said that police would look at charges including wearing the uniform of an organisation that perpetrated crimes against humanity and making racist slurs.
Last year fashion designer John Galliano was convicted of public insult and antisemitism under French anti-racism laws.
Ian Austin, a Labour MP and the adopted son of a Jewish man who fled Nazi Germany, said that the formal investigation showed that it was time for the Conservative Party to withdraw the whip from Mr Burley.
Prime Minister David Cameron asked for a fuller investigation of the matter in Britain, but so far the results have not been made public.
A Conservative Party spokesman declined to comment on whether the whip would be withdrawn, but he said that a summary of the findings of the investigation would be published by the end of February. He said the findings would be made available to all interested parties.
