WASHINGTON—Senior White House officials are discussing plans to pursue an overhaul of how the government surveils individuals in the U.S. suspected of posing a national security risk, spurred in part by President Trump’s grievances about an investigation of an adviser on his 2016 presidential campaign, 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…

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