Federal prosecutors in New York are investigating whether Rudy Giuliani stood to profit personally from a Ukrainian natural-gas business pushed by two associates who also aided his efforts there to launch investigations that could benefit President Trump, people familiar with the matter said.

Mr. Giuliani’s associates, Lev Parnas and Igor Fruman, pitched their new company, and plans for a Poland-to-Ukraine pipeline carrying U.S. natural gas, in meetings with Ukrainian officials and energy executives this year, saying the project...