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.
Mr. Flynn lasted only 24 days as national security adviser, but his legal troubles now lie at the center of a political storm that has engulfed the Trump administration. For months, prosecutors have used multiple grand juries to issue subpoenas for documents related to Mr. Flynn.
President Trump has publicly said Mr. Mueller should confine his investigation to the narrow issue of Russia’s attempts to disrupt last year’s presidential campaign, not conduct an expansive inquiry into the finances of Mr. Trump or his associates.
Mr. Flynn declined to comment. Ty Cobb, special counsel to Mr. Trump, said, “We’ve said before we’re collaborating with the special counsel on an ongoing basis.”
“It’s full cooperation mode as far as we are concerned,” he said.
After Mr. Flynn’s dismissal, Mr. Trump tried to get James B. Comey, the F.B.I. director, to drop the investigation, Mr. Comey said.
Mr. Mueller is investigating whether Mr. Trump committed obstruction of justice in pressing for an end to the Flynn inquiry. The president fired Mr. Comey on May 9.
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.
They have also asked about the White Canvas Group, a data-mining company that was reportedly paid $200,000 by the Trump campaign for unspecified services. The Flynn Intel Group shared office space with White Canvas Group, which was founded by a former special operations officer who was a friend of Mr. Flynn’s.
Mr. Flynn has now had to file three versions of his financial-disclosure forms. His first version did not disclose payments from Russia-linked companies. He added those payments to an amended version of the forms he submitted in March. This week he filed a new version, adding that he briefly had a contract with SCL Group, the parent company of Cambridge Analytica, a data-mining firm that worked with the Trump campaign.
The new forms list at least $1.8 million in income, up from roughly the $1.4 million he had previously reported. It is unclear how much of that money was related to work Mr. Flynn did on Turkey issues.
Mr. Flynn’s campaign to discredit the opponent of the Turkish government, Fethullah Gulen, began on Aug. 9 when his firm signed a $600,000 deal with Inovo BV, a Dutch company owned by Mr. Alptekin, a Turkish-American businessman.
Mr. Gulen, a reclusive cleric, lives in rural Pennsylvania.
The contract with Mr. Alptekin was brought in by Bijan R. Kian, an Iranian-American businessman who was one Mr. Flynn’s business partners. Mr. Kian, who served until 2011 as a director of the Export-Import Bank, a United States federal agency, is also under scrutiny, according to witnesses questioned by Mr. Mueller’s investigators. A lawyer for Mr. Kian declined to comment.
Inovo ultimately paid the Flynn Intel Group only $530,000 and received little more than slapdash research and a comically inept attempt to make an anti-Gulen video, which was never completed. The entire enterprise would probably have gone unnoticed if Mr. Flynn had not written an opinion piece advocating improved relations between Turkey and the United States and calling Mr. Gulen “a shady Islamic mullah.”
The opinion piece appeared on Election Day. Soon after, The Daily Caller revealed that the Flynn Intel Group had a contract with Inovo, prompting the Justice Department look into Mr. Flynn’s relationship with Mr. Alptekin.
The authorities quickly determined that Mr. Flynn had not registered as a foreign agent, as required by law. In March, he retroactively registered with the Justice Department.
Mr. Mueller’s investigators have asked repeatedly about two payments of $40,000 each that the Flynn Intel Group made to Inovo, said witnesses who have been interviewed in the case.
The investigators have indicated that they suspect that the payments were kickbacks, and in one interview pointed to the suspicious timing of the transfers. The first payment back to Inovo was made on Sept. 13, four days after the Dutch company made it first payout under the contract, sending $200,000 to the Flynn Intel Group.
On Oct. 11, Inovo paid the Flynn Intel Group an additional $185,000. Then, six days later, Flynn Intel Group sent $40,000 to Inovo.
Mr. Alptekin said that both payments were refunds for work that the Flynn Intel Group had not completed.
“Ekim maintains that all payments and refunds were for unfulfilled work, and that they were legal, ethical and above board,” said Molly Toomey, a spokeswoman for Mr. Alptekin. She described the reimbursements as “a business decision.”
Another focus for investigators is the repeatedly changing explanation Mr. Alptekin has offered for why he hired Mr. Flynn. In March, he told a reporter that Mr. Flynn had been hired “to produce geopolitical analysis on Turkey and the region” for an Israeli energy company. But in an interview with The New York Times in June, he said he wanted a credible American firm to help discredit Mr. Gulen, whom the President Recep Tayyip Erdogan of Turkey has blamed for the coup attempt.
“Like many Americans rolling up their sleeves in 9/11 to do something, I decided to do something,” Mr. Alptekin said.
He scoffed at the suggestion that he was a front for the Turkish government. Inovo, he noted, was registered in the Netherlands, where it is difficult to mask the ownership of a company. A clear paper trail linked the payments between his company and the Flynn Intel Group, he said.
“If we were trying to hide,” he said, “you’d think we’d be good at it.”