Leave.EU was a separate campaign to the official Vote Leave organisation.
Without mentioning allegations of Russian money or interference in the campaign, the commission said it had reason to believe a number of criminal offenses may have been committed.
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"Our investigation has unveiled evidence that suggests criminal offences have been committed which fall beyond the remit of the Commission. Now Aaron Banks is under investigation because Leave.EU stands accused of spending foreign money", Lammy said on Twitter.
She denied claims the money had come from Russian sources, saying: "I run the group of companies where the money was from and we don't have any transactions that are from Russia".
Arron Banks blamed "intense political pressure from anti-Brexit supporters" for the decision by the National Crime Agency to open an investigation.
Banks said in a statement that he welcomes the investigation and is confident it will debunk the "ludicrous allegations" against him. I am confident that a full and frank investigation will finally put an end to the ludicrous allegations levelled against me and my colleagues.
It went on to allege that, in fact, Mr Banks was "not the true source of the £8m loans" and the money instead came from "Rock Holdings Limited, which is incorporated in the Isle of Man". The Electoral Commission has produced no evidence to the contrary.
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He added: "There is no evidence of any wrongdoing from the companies that I own".
"Any thought that the electoral commission are neutral in what they do is, I think, completely thrown out of the window by looking at the composition of the them, of the 10 people who are on that board".
He promised he had not received any foreign money and that he is a United Kingdom taxpayer.
Arron Banks, Elizabeth Bilney and others involved in Better for the Country, Leave.EU and associated companies concealed the true details of these financial transactions.
Bob Posner, the Electoral Commission's director of political finance and regulation and legal counsel, said the the claims were "significant" because at least £2.9m of the money referred to was money was "used to fund referendum spending and donations during the regulated period of the European Union referendum" - when funding restrictions are in place.
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