Behind closed doors, committee and staff members have been researching the law, parliamentary rules and past precedents as they discuss whether to proceed, people familiar with the inquiry said.

Mike Stern, a former senior counsel to the House who has not been involved in the talks, said he had little doubt the committee had the legal authority to issue such a subpoena. The hurdle, he suggested, is the prospect of further shredding congressional norms of comity.

“People have been asking me whether Congress has the power to subpoena their members, and I think the answer is clearly yes, and I haven’t really heard any argument to the contrary,” said Mr. Stern, who served under Republican leaders. “In the context of the highly partisan political fight here, it’s just a very heavy lift.”