WASHINGTON — As the special counsel’s investigators pursued the question of whether President Trump tried to impede their work, they uncovered compelling evidence — a voice mail recording and statements from a trusted witness — that might have led to him.

A lawyer for Mr. Trump, John M. Dowd, reached out to a lawyer for a key witness who had just decided to cooperate with the government, Michael T. Flynn. Mr. Dowd fished in his message for a heads-up if Mr. Flynn was telling investigators negative information about Mr. Trump — while also appearing to say that if Mr. Flynn was just cutting a deal without also flipping on the president, then he should know Mr. Trump still liked him.

But the president’s role, if any, remains a mystery. Mr. Dowd never said whether Mr. Trump directed him to make the overture. And investigators for the special counsel, Robert S. Mueller III, declined to question Mr. Dowd about his message, citing “attorney-client-privilege issues.”

The release of the recording last week served as a reminder that despite the considerable evidence laid out in the 448-page Mueller report, some tantalizing questions about the president’s conduct went unanswered because investigators encountered obstacles or backed off on pursuing leads.