House Republican leaders are weighing “further steps” to force former top White House strategist Stephen K. Bannon to answer investigators’ questions in their probe of Russian meddling in the 2016 election — including potentially declaring him in contempt of Congress — after a Thursday interview they called “frustrating.”

Bannon came to speak with the House Intelligence Committee under a subpoena the panel issued on the spot last month, when he refused to answer questions related to the transition period and his tenure in the White House. The interview came after Bannon met with investigators in special counsel Robert S. Mueller III’s probe on Monday and Wednesday, according to a person familiar with the sessions.

Bannon has put no preconditions on his interviews with Mueller. But he presented intelligence panel members with a list of only 25 questions that he would be willing to answer related to anything that took place after Donald Trump won the 2016 election. According to the panel’s top Democrat, Rep. Adam B. Schiff (Calif.), those questions had been “literally scripted” by the White House, and Bannon’s answer to all of them was “no.”

When the committee tried to push Bannon to answer questions that were not on his list, he repeatedly told members that the White House had not authorized him to engage on those queries. At no point, people familiar with the interview said, did Bannon voluntarily elaborate on his answers.

The intelligence panel’s probe is not supposed to overlap with the objectives of Mueller’s investigation, but several events and people are common to both efforts. Bannon has not yet met with the Senate Intelligence Committee in its probe of Russian meddling in the election.

But in the House, Republicans and Democrats alike have been angered by Bannon’s repeated attempts to dismiss questions based on a claim to executive privilege that Trump never formally invoked, even when served with a subpoena.

Intelligence Committee member K. Michael Conaway (R-Tex.) said Thursday that he, House Speaker Paul D. Ryan (R-Wis.) and a few others would decide whether to accept Bannon’s legal arguments against answering the panel’s questions or take punitive measures such as declaring him in contempt. The decision-makers will not include panel chairman Devin Nunes (R-Calif.), Conaway said.

When asked whether he was ready to issue a contempt citation for Bannon, Conaway said only: “I think he should answer our questions.”

Republican leaders are not expected to decide on a course of action until late February, after they return to Washington following a one-week recess, he said.

Schiff, however, demanded that the committee move to hold Bannon in contempt as soon as possible.

“I think contempt is the only road left open to us,” the Democrat said.

Bannon’s return to the committee was scheduled and delayed three times while the White House hammered out the terms of the interview with the House counsel. On Wednesday night, the White House sent the committee a letter outlining its argument for why executive privilege could apply to the transition period, according to panel members. But lawmakers said that letter was not a formal invocation of executive privilege, and they continue to reject the premise that privilege can apply to the period when Trump was not in the Oval Office.

Panel members on both sides of the aisle also stressed that Bannon could not cite nonexistent privilege as an excuse to avoid their questions.

“That’s not how privilege works,” Schiff said. “That’s how stonewalling works.”

It is not clear what type of contempt the committee could seek to declare against Bannon. Should lawmakers seek a citation, a vote in the House Intelligence Committee would first be required — and later, probably, a resolution by the whole House — before the case would be transferred to the courts.

If Bannon does not settle with the committee, the matter could linger in the courts far beyond the panel’s projected schedule to wrap up its probe. That, Schiff surmised, could be “part of the White House stratagem.”

“They have decided they want to stop our committee, and they hope to draw this out long enough,” Schiff said.

The House panel’s probe has long been plagued by partisan divisions. But Bannon’s fight with the committee has drawn Democrats and Republicans together in a rare common cause.

Rep. Trey Gowdy (R-S.C.) led the push for Bannon to answer lawmakers’ questions and to issue him a subpoena. Now several Republicans say that holding Bannon in contempt, if he does not cooperate, will be necessary to send a message to this and future administrations that they cannot ignore congressional oversight.

“If you don’t, I mean, what kind of precedent is that sending? For not just our committee, but every committee?” Rep. Thomas J. Rooney (R-Fla.), who was deputized to help run the committee’s Russia probe, said Wednesday. He said panel and party leaders would be likely to sign off on the move.

Rosalind S. Helderman contributed to this report.