The entrepreneur will continue leading Donald J. Trump’s government efficiency project before turning his attention to a campaign. He has encountered some turbulence with Mr. Trump, and with Elon Musk.

Donald Trump’s ally, Vivek Ramaswamy, was bypassed for a Senate appointment in his home state of Ohio despite encouragement by the president-elect, foreclosing a political possibility for the entrepreneur at the start of the new administration.

The co-leader of Trump’s Department of Government Efficiency initiative plans to announce a run for Ohio governor, two people familiar with his plans said.