In one episode, the president told an aide that the White House counsel, Donald F. McGahn II, should issue a statement denying a New York Times article in January. The article said Mr. McGahn told investigators that the president once asked him to fire the special counsel, Robert S. Mueller III. Mr. McGahn never released a statement and later had to remind the president that he had indeed asked Mr. McGahn to see that Mr. Mueller was dismissed, the people said.

In the other episode, Mr. Trump asked his former chief of staff, Reince Priebus, how his interview had gone with the special counsel’s investigators and whether they had been “nice,” according to two people familiar with the discussion.

The episodes demonstrate that even as the special counsel investigation appears to be intensifying, the president has ignored his lawyers’ advice to avoid doing anything publicly or privately that could create the appearance of interfering with it.