Lawyers for Mr. Flynn and Mr. Trump declined to comment. The four people briefed on the matter spoke on the condition of anonymity because they were not authorized to discuss the matter publicly.
A deal with Mr. Flynn would give Mr. Mueller a behind-the-scenes look at the Trump campaign and the early tumultuous weeks of the administration. Mr. Flynn was an early and important adviser to Mr. Trump, an architect of Mr. Trump’s populist “America first” platform and an advocate of closer ties with Russia.
His ties to Russia predated the campaign — he sat with President Vladimir V. Putin at a 2015 event in Moscow — and he was a point person on the transition team for dealing with Russia.
Among the interactions that Mr. Mueller is investigating is a private meeting that Mr. Flynn had with the Russian ambassador and Jared Kushner, the president’s son-in-law, during the presidential transition. In the past year, it has been revealed that the Trump campaign repeatedly tried to meet with Russian officials who were promising compromising information on Mrs. Clinton.
Mr. Flynn is regarded as loyal to Mr. Trump, but he has in recent weeks expressed serious concerns to friends that prosecutors will bring charges against his son, Michael Flynn Jr., who served as his father’s chief of staff and was a part of several financial deals involving the elder Mr. Flynn that Mr. Mueller is scrutinizing.
The White House has said that neither Mr. Flynn nor other former aides has incriminating information to provide about Mr. Trump. “He likes General Flynn personally, but understands that they have their own path with the special counsel,” a White House lawyer, Ty Cobb, said in an interview last month with The New York Times. “I think he would be sad for them, as a friend and a former colleague, if the process results in punishment or indictments. But to the extent that that happens, that’s beyond his control.”