WASHINGTON—The Justice Department asked its internal watchdog to examine if there was any impropriety in the counterintelligence investigation of President Donald Trump’s 2016 campaign, after the president demanded Sunday that the department investigate the motives behind the inquiry.

Earlier Sunday, in one of a series of tweets targeting the probe into whether Trump associates colluded with Russia during the 2016 campaign, Mr. Trump wrote: “I hereby demand, and will do so officially tomorrow, that the Department of Justice...