Mr. Rodriguez and Ms. Haspel were later investigated by John Durham, an assistant United States attorney. Mr. Durham ultimately recommended filing no charges over the tape destruction, but his report laying out his findings and reasoning is secret. (The New York Times lost a Freedom of Information Act lawsuit to make it public.)

The Trump administration has disclosed the executive summary of Mr. Durham’s report to the Intelligence Committee, and on Thursday, Senator Jeff Flake, Republican of Arizona, wrote to Attorney General Jeff Sessions asking for it to be made available to the full Senate, calling it “critically important” that all lawmakers have access to it. Several Democratic senators had already asked for that step.

Also on Thursday, Senator Ron Wyden, Democrat of Oregon, said he wanted to know more about an apparent discrepancy in how Ms. Haspel and Mr. Rodriguez have described their discussions about the tape destruction order.

In an interview published on Wednesday by ProPublica, Mr. Rodriguez said he told Ms. Haspel that he was going to take matters into his own hands because he had concluded that the director at the time, Porter Goss, would never approve their destruction. But Ms. Haspel said on Wednesday that the conversation never happened. She said that Mr. Rodriguez instead told her that he would discuss the issue with Mr. Goss before issuing the order, but then he sent it without having spoken to Mr. Goss.

Mr. Rodriguez’s comments were ambiguous, however. He also told ProPublica, “She may have thought I was going to talk to more people about it before hitting ‘send,’ but I had made up my mind.” But Mr. Wyden said that the Senate needed to clarify what happened.

“Gina Haspel has claimed for years that she believed her boss would get signoff before ordering the destruction of the torture tapes,” Mr. Wyden said. “But her boss says he told her otherwise. We need to get to the bottom of what Gina Haspel did, and what she knew, before the Senate votes on her nomination.”

Ms. Haspel seems likely to be narrowly confirmed. One Republican senator, John McCain of Arizona, said he would oppose her, but a Democrat, Joe Manchin III of West Virginia, said he would support her. The Intelligence Committee is likely to receive written answers to its follow-up questions early next week, after which the panel would vote, leading to a full Senate vote late next week or, more likely, early the following one, congressional aides said.