The data demand stems from legal action by Prof Carroll, who wants to know what information the firm holds on him.

The company is at the centre of a row over the way it grabbed data on millions of Facebook users.

Cambridge Analytica could face a steep fine if it does not comply before a 30-day deadline expires.

Political views

Prof Carroll - an associate professor at Parsons School of Design in New York - was prompted to find out what information it had gathered about him when it emerged that Cambridge Analytica had built up profiles of 240 million Americans.

He sent a data request to Cambridge Analytica. In March last year, he got back information that showed how it had scored him on a small set of political categories, including gun control and national security.

Prof Carroll then launched legal action to find out more, as he believed the information sent to him was incomplete. Cambridge Analytica had previously boasted that every voter profile it generated used 4,000 to 5,000 data points.

As the company named as Cambridge Analytica's data controller, SCL Elections, was based in the UK, Prof Carroll launched legal action at the High Court in London and also filed a complaint with the UK Information Commissioner's Office.

"This should solve a lot of mysteries about what the company did with data and where it got it from," he told the Guardian newspaper.

The High Court case is due to be heard in the next few months.

 In early May, work began on shutting down Cambridge Analytica and SCL Elections

In a letter sent to the firm, the Information Commissioner said it wanted to know where the data on Mr Carroll came from and what had been done with it.

"The company has consistently refused to co-operate with our investigation into this case and has refused to answer our specific enquiries in relation to the complainant's personal data," Information Commissioner Elizabeth Denham said in a statement.

Cambridge Analytica, which has repeatedly said it did nothing wrong in the way it processed data, is under fire for allegedly using the personal information of millions of Facebook users for political campaigning, without their consent or knowledge.

Neither Cambridge Analytica nor SCL Elections have responded to a request for comment on the ICO demand.

Last week, bankruptcy proceedings were started for Cambridge Analytica and SCL Elections. In a statement, the blame for the closure was put on a "siege of media coverage" that had hit the two companies in the wake of the Facebook data-harvesting scandal.