The April 11 letter from the inspector general did not specify what payment from a foreign government was at issue. But Mr. Flynn is known to have been paid more than $45,000 by RT, the Kremlin-financed news network, to give a speech in Moscow in December 2015.

More than two months after Mr. Flynn was fired as national security adviser, his links to Moscow continue to bedevil the Trump administration, which itself has been dogged by reports of ties between Russian officials and the president’s associates. The Senate and House Intelligence Committees and the F.B.I. are investigating allegations that Mr. Trump’s associates colluded with Russia before and after last year’s presidential election.

Mr. Flynn, an unconventional and often contentious former three-star Army general, is at the center of the inquiries. Apart from the paid speech in Moscow — a trip during which he was photographed dining at the elbow of President Vladimir V. Putin of Russia — there are questions about conversations he had last year with the Russian ambassador to the United States. He was fired for misleading Vice President Mike Pence and other top officials about those conversations.

It is not clear whether Mr. Flynn is cooperating with any of the investigations. But last month, he offered to sit down with congressional investigators in exchange for immunity, a deal that lawmakers say was declined, at least for now.

The White House has so far declined requests from both congressional Republicans and Democrats to provide the House Oversight Committee with internal documents related to Mr. Flynn.