Ever since Britons voted to quit the European Union in 2016, questions have been asked about Arron Banks, a pugnacious entrepreneur with wide-ranging international contacts, who bankrolled a pro-withdrawal campaign and is thought to be the biggest political donor in British political history.

On Thursday, the police were investigating if Mr. Banks broke the law while promoting Britain’s departure, known as Brexit.

The election law watchdog, the Electoral Commission, said it had referred the matter to the National Crime Agency, the police who tackle organized crime, because it had reasonable grounds to think that “a number of criminal offenses may have been committed.”

Mr. Banks, sometimes described as the “godfather of Brexit,” denied any wrongdoing and predicted that the investigation would “finally put an end to the ludicrous allegations leveled against me.”