Outgoing Central Intelligence Agency Director John Brennan on Monday rejected Donald Trump’s suggestions that he may have leaked an unsubstantiated dossier on the president-elect while defending the U.S. intelligence community more broadly from Mr. Trump’s recent attacks on its credibility and integrity.

Mr. Brennan said he had no way to assess the allegations contained in the dossier of political opposition research, which included inflammatory personal information and suggestions of contacts between Mr. Trump and the Russian...