Investigators working for the special counsel, Robert S. Mueller III, recently asked the White House for documents related to the former national security adviser Michael T. Flynn, and have questioned witnesses about whether he was secretly paid by the Turkish government during the final months of the presidential campaign, according to people close to the investigation.
Though not a formal subpoena, the document request is the first known instance of Mr. Mueller’s team asking the White House to hand over records.
The company was paid $530,000 to run a campaign to discredit an opponent of the Turkish government who has been accused of orchestrating last year’s failed coup in the country.
Investigators want to know if the Turkish government was behind those payments — and if the Flynn Intel Group made kickbacks to the businessman, Ekim Alptekin, for helping conceal the source of the money.
The line of questioning shows that Mr. Mueller’s inquiry has expanded into a full-fledged examination of Mr. Flynn’s financial dealings, beyond the relatively narrow question of whether he failed to register as a foreign agent or lied about his conversations and business arrangements with Russian officials.
Investigators are also examining the flow of money into and out of the Flynn Intel Group — a consulting firm Mr. Flynn founded after being forced out as the director of the Defense Intelligence Agency — according to several potential witnesses who have been interviewed by prosecutors and F.B.I. agents.
Taking money from Turkey or any foreign government is not illegal. But failing to register as a foreign agent is a felony, and trying to hide the source of the money by routing it through a private company or some other entity, and then paying kickbacks to the middleman, could lead to numerous criminal charges, including fraud.
Prosecutors have also asked during interviews about Mr. Flynn’s speaking engagements for Russian companies, for which he was paid more than $65,000 in 2015, and about his company’s clients — including work it may have done with the Japanese government.