Things are very much heating up on this. I don’t have an answer on the path to the floor for a vote just yet, but it’s accelerating,” Magaziner said.

The speaker has repeatedly lost his grip on the House floor thanks to a once rare parliamentary maneuver that G.O.P. members are increasingly using to force action on legislation.

A bipartisan group in Congress is advancing a bill to ban lawmakers from buying or selling individual stocks and to require ...