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 the White Canvas Group, which was founded by a former Special Operations officer who was a friend of Mr. Flynn’s.