WASHINGTON—Congress is probing the Obama administration’s decision to use research by an ex-British spy to justify, in part, surveillance of an associate of​ Donald Trump, as Republican scrutiny rises of law-enforcement actions during 2016 presidential campaign. In the final days of the campaign, the Justice Department used information from Christopher Steele, a former British intelligence official, as part of its request for a secret court order to monitor Carter Page, a foreign policy adviser to Mr. Trump and his ties to Russia, say people familiar with the matter.