Private-sector backers of a controversial Middle East nuclear-power plan worked with former national security adviser Mike Flynn to promote it inside the White House, to the point of sending him a draft memo for the president to sign authorizing the project.

At issue was a proposal to build dozens of nuclear reactors, billed by its backers as a “Marshall Plan for the Middle East.” Before joining the White House, Mr. Flynn, a retired lieutenant general, had advised some of the U.S. companies involved in the plan in his capacity...