Dozens of Senate Republicans said Wednesday they will formally challenge President Biden’s COVID-19 vaccine rules on large employers by filing a resolution that would rely on congressional review powers to strike it down.

Sen. Mike Braun of Indiana led 40 party colleagues in moving to “disapprove and nullify” the forthcoming rule from the Occupational Safety and Health Administration, which will be published any day now and would require companies with 100 or more workers to mandate the vaccine or weekly testing.

Republicans have decried the regulation, which Mr. Biden announced in September to lift vaccination rates, as an unacceptable intrusion on personal medical decisions.

Mr. Braun’s push relies on the Congressional Review Act, which allows Congress to overturn certain federal regulations if both chambers pass a joint resolution of disapproval and the president signs it.

“Today, we are one step closer to protecting the liberties of millions of Americans in the private sector workforce under the Congressional Review Act. I urge my Senate colleagues to vote in favor of this disapproval resolution in the coming weeks,” Mr. Braun said.

With Democrats unlikely to join the effort and Mr. Biden not set to disapprove his own actions, the main impact of the effort will be to put Democrats on the record in a vote to support or reject Mr. Biden’s mandate after a 20-day review period following publication of the OSHA rule in the Federal Register.

The mandate has sparked controversy since Mr. Biden floated it as part of a broader set of mandates on federal employees and contractors.