WASHINGTON—Senior White House officials are discussing an overhaul of the government’s surveillance program for people in the U.S. suspected of posing a national-security risk, spurred in part by President Trump’s grievances about an investigation of a 2016 campaign adviser, according to people familiar with the matter.

The effort seeks to take advantage of the looming expiration of some spying powers next month, including portions of the Foreign Intelligence Surveillance Act, a Watergate-era law that Mr. Trump believes was...