Confident that a Biden administration was on the horizon, Mr. Schmidt, a swaggering former political adviser to John McCain and Arnold Schwarzenegger, pitched the other attendees on his post-Trump vision for the project over a breakfast of bagels and muffins. And it was ambitious.

“Five years from now, there will be a dozen billion-dollar media companies that don’t exist today,” he told the group, according to two people who attended. “I would like to build one, and would invite all of you to be part of that.”

In fact, Mr. Schmidt and the three other men who started the Lincoln Project — John Weaver, Reed Galen and Rick Wilson — had already quietly moved to set themselves up in the new enterprise, drafting and filing papers to create TLP Media in September and October, records show. Its aim was to transform the original project, a super PAC, into a far more lucrative venture under their control.

This was not the only private financial arrangement among the four men. Shortly after they created the group in late 2019, they had agreed to pay themselves millions of dollars in management fees, three people with knowledge of the deal said.