WASHINGTON—Howard Lutnick’s journey from Wall Street veteran to one of President Trump’s most hawkish trade advisers began with a speech at an October campaign rally at Madison Square Garden.

Lutnick extolled the virtues of tariffs and argued that the duties could at least in part replace the U.S. income tax, a line that Trump has regularly used as a talking point. Since then, Lutnick has privately built an alliance with two of Trump’s most vital first-term trade advisers and proponents of punitive tariffs: Peter Navarro and Jamieson Greer, according to people familiar with the matter.