Mr. Hogan, who is trying to play a larger role in national politics as his second term as governor of a solidly Democratic state draws to an end in 2022, will join former Senator Joseph I. Lieberman, Democrat of Connecticut, in leading the organization, which supports the 50-member House Problem Solvers Caucus and a smaller group of eight centrist senators.

Those lawmakers in recent weeks have been working on a stimulus compromise to break the gridlock over coronavirus relief.

“No Labels is really at a great moment right now of realizing the goals of the organization when it was started 10 years ago,” Mr. Lieberman said Monday. “I expect Governor Hogan will be expanding our reach to the nation’s governors.”