The National Crime Agency is to investigate allegations of multiple criminal offences by Arron Banks and his unofficial leave campaign in the Brexit referendum, prompting calls from some MPs for the process of departing the European Union to be suspended.

The NCA would look into suspicions that a “number of criminal offences may have been committed”, the Electoral Commission said in a statement, saying there were reasonable grounds to suspect Banks was “not the true source” of £8m in funding to the Leave.EU campaign.

The commission said the cases involve Banks, the insurance millionaire who heavily backed leave; Elizabeth Bilney, one of his key associates; Leave.EU itself; the company used to finance it; and “other associated companies and individuals”.

News of the investigation prompted anti-Brexit campaigners to call for a delay to the process of leaving the EU. The Labour MP David Lammy said Brexit “must be put on hold until we know the extent of these crimes against our democracy”.

A series of other Labour MPs echoed the call, while the Lib Dems said Brexit could not go ahead based on “a leave campaign littered with lies, deceit and allegations of much worse”.

Downing Street said it could not comment on a live investigation, but dismissed the idea of a pause: “The referendum was the largest democratic exercise in this country’s history and the PM is getting on with delivering its result.”

Banks and Bilney, who chaired the Leave.EU campaign, said they rejected any allegations of wrongdoing, and argued the investigation was motivated by political considerations.

The commission said the investigations would focus on £2m reported to have been lent to Better for the Country (BFTC), a company that was used to finance Leave.EU, and £6m more reportedly provided to the organisation, on behalf of Leave.EU, by Banks.

Of the total, £2.9m was used to fund referendum spending on behalf of Leave.EU.

A statement said: “Due to multiple suspected offences, some of which fall outside the commission’s remit, the commission has referred this matter and handed its evidence to the National Crime Agency.”

The NCA confirmed it had begun an investigation connected to “suspected electoral law offences covered by that referral, as well as any associated offences”.

It said: “While electoral law offences would not routinely fall within the NCA’s remit, the nature of the necessary inquiries and the potential for offences to have been committed other than under electoral law led us to consider an NCA investigation appropriate in this instance.”

Banks, formerly a major bankroller of Ukip, said in a statement he was pleased at the news. He said: “I am confident that a full and frank investigation will finally put an end to the ludicrous allegations levelled against me and my colleagues.

“There is no evidence of any wrongdoing from the companies I own. I am a UK taxpayer and I have never received any foreign donations. The Electoral Commission has produced no evidence to the contrary. The Electoral Commission has referred me to the National Crime Agency under intense political pressure from anti-Brexit supporters.”

Bilney told BBC Radio 4’s World at One on Thursday that the Electoral Commission had consistently taken “a biased approach”, saying this was owing to its previous funding from the EU.

A series of reports, led by the Observer’s Carol Cadwalladr, have uncovered links by Banks and his leave campaign to Russia, prompting speculation about this being the source of some of the funding.

But Bilney rejected the idea, telling the BBC: “I can confirm it wouldn’t have come from Russia … I run the group companies where the money was from and we don’t have any transactions that are from Russia.”

Theresa May’s spokeswoman said that to date the government had “not seen successful interference in UK democratic processes”.

She added: “We of course remain vigilant and we will continue to work to strengthen our democracy against potential interference.”

The investigation centres on allegations the £8m provided to the campaign involved money from a “non-qualifying or impermissible company” based in the Isle of Man, and that Banks, Bilney and others involved in BFTC, Leave.EU and others sought to cover up the origin of the funding.

Bob Posner, the Electoral Commission’s director of political finance and regulation, said: “We have reasonable grounds to suspect money given to Better for the Country came from impermissible sources and that Mr Banks and Ms Bilney, the responsible person for Leave.EU, knowingly concealed the true circumstances under which this money was provided.

“This is significant because at least £2.9m of this money was used to fund referendum spending and donations during the regulated period of the EU referendum.

“Our investigation has unveiled evidence that suggests criminal offences have been committed which fall beyond the remit of the commission. This is why we have handed our evidence to the NCA to allow them to investigate and take any appropriate law enforcement action. This is now a criminal investigation.”

In a report on the investigation published alongside its statement, the commission said BFTC spent at least £2.9m of the money in question – £750,000 on payments for Leave.EU and £2.2m donated to other campaigners. Both Leave.EU and BFTC had said all the money came only from Banks, it added.

However, the investigations found parties involved in the transactions included Rock Holdings Limited, a company majority-owned by Banks, which is incorporated in the Isle of Man. Under electoral law, companies based outside the UK are not allowed to donate or loan to political campaigns.

When asked about the circumstances, Banks and Bilney “gave us unsatisfactory explanations about these transactions, and we have reasonable grounds to suspect that they knowingly concealed and sought to conceal the true circumstances”, the report said.

Of the money provided to other organisations, the report said, £1.9m was donated to Grassroots Out, with smaller sums going to Trade Unionists Against the EU, Ukip, Veterans for Britain, and the TV station Wag TV, which made a pro-Brexit documentary.