President Donald Trump’s personal lawyer negotiated a deal in late 2017 to pay $1.6 million to a former Playboy model who said she was impregnated by a top Republican fundraiser, according to people familiar with the matter.

Michael Cohen, whose office, home and hotel room were raided by federal agents this week, arranged the payments to the woman on behalf of Elliott Broidy, a deputy finance chairman of the Republican National Committee with ties to Mr. Trump, the people familiar with the matter said. Mr. Broidy, a Los Angeles-based...