MPs looking into fake news have issued a formal summons to both ex-Cambridge Analytica boss Alexander Nix and Dominic Cummings, former campaign director of Vote Leave.
They will appear before the Digital, Culture, Media and Sport committee,
They had each said that they could not appear while investigations by the Information Commissioner’s Office and the Electoral Commission were ongoing.
But both bodies confirmed appearing would not hinder their inquiries.
The MPs are looking at how consulting firm Cambridge Analytica harvested the data of millions of Facebook users worldwide.
Damian Collins MP, chairman of the committee, said: “We are summoning Mr Nix to Parliament to get to the truth about an extremely serious issue affecting over one million UK Facebook users, and potentially voters in elections worldwide.
“Mr Nix has been given multiple opportunities to appear and clarify his evidence to the committee, but now we expect him to appear on 6 June.
“There are serious inconsistencies between Mr Nix’s original testimony of 27 February, and evidence received under the inquiry since.”
London-based Cambridge Analytica – which announced it was closing earlier this month – is accused of acquiring data from up to 87 million Facebook profiles for use in political campaigns. The firm has denied any wrongdoing.
There are also questions about links between Cambridge Analytica and the Vote Leave campaign.
Vote Leave spent £2.7m on the services of Canadian digital agency AIQ in the run-up to the June 2016 EU referendum and the firm has admitted that it also conducted work for Cambridge Analytica’s parent firm SCL.
Mr Collins said: “We hoped that Mr Cummings would have responded positively to our requests for him to appear, considering the allegations made against the Vote Leave campaign during our inquiry.”
He added: “22 May will be his opportunity to clarify allegations about the unlawful coordination of EU referendum campaigns, campaign spending, and misuse of people’s personal data.”
If either of the men fails to attend, the committee can report the matter to Parliament.
“This could result in a decision that a contempt of Parliament has been committed,” Mr Collins said.
The MPs are also keen to interrogate Facebook founder Mark Zuckerberg, after he declined a request to appear, sending instead Mike Schroepfer, the social network’s chief technology officer.
MPs deemed that he failed to answer 40 of their questions.
The committee is due to hear evidence from Jeff Silvester, from AIQ, next week.
According to Facebook’s testimony to MPs, AIQ spent $2m (£1.5m) on advertising on its platform during the Brexit referendum.
In separate news, The Fair Vote Project, which was set up to represent those whose data was harvested, is preparing a damages claim against Facebook.
If successful, the social network could faces damages running into billions of pounds.