The calls between Mr. Flynn and Mr. Kislyak were referenced repeatedly in court documents and the special counsel’s report on Russian election interference but never released, and prosecutors have not acknowledged the existence of the wiretap. Judge Sullivan, who is overseeing Mr. Flynn’s case, ordered that audio recordings of his conversations with Mr. Kislyak be made public along with a voice mail message made by the president’s lawyer.

The Justice Department’s refusal to comply with the judge’s order made clear that prosecutors had no interest in confirming the wiretap, which was approved by the secretive Foreign Intelligence Surveillance Court.

“This would be a rare step to make public” such intelligence collection, said Joshua Geltzer, a former Justice Department official. “What you see in today’s filing is the government trying to avoid disclosing that material.”