Carter Page, who served as a foreign-policy adviser to Donald Trump’s campaign, was known to U.S. counterintelligence officials for years before he became a prominent figure in a dossier of unverified research about the future president’s ties to Russia.

The White House is expected to release as early as this week a memo detailing what Republicans allege were surveillance abuses during the 2016 campaign. Republicans say the memo, written by the GOP staff on the House Intelligence Committee, shows that prosecutors used information...