WASHINGTON — President Trump’s assertion that a senior law enforcement official privately gave assurances that he is not under scrutiny in the investigations shadowing him partly served as a defense strategy but also raised new questions about how the Justice Department is conducting the inquiries.

Mr. Trump said in an interview on Thursday with The New York Times that the deputy attorney general, Rod J. Rosenstein, has told him that he was not a target of the special counsel investigation into Russia’s election interference and whether any Trump associates conspired with it. “Rod told me I’m not a target of the investigation,” he said. The president also said that Mr. Rosenstein reiterated the statement to his personal lawyers “a number of times,” but did not say when.

Questions about how law enforcement officials are navigating the inquiry have hung over the Justice Department since a separate investigation came into view in April, when F.B.I. agents raided the office, home and hotel room of Mr. Trump’s former longtime lawyer and fixer, Michael D. Cohen. The raids prompted Mr. Trump to increase his attacks on the Justice Department, raising fears among Democrats that he might shut down that inquiry or the one by the special counsel, Robert S. Mueller III.

But Mr. Rosenstein told the president shortly after the raid that he was not a target in either investigation, according to a Bloomberg News report citing unnamed people familiar with the matter, which prompted speculation that Mr. Rosenstein was trying to placate the president to allow the investigations to continue. People close to Mr. Trump’s legal team confirmed that account.