WASHINGTON — The chairman of the House committee investigating the Jan. 6 attack on the Capitol set a vote next week to recommend that Stephen K. Bannon, a former top adviser to President Donald J. Trump, face criminal contempt charges for refusing to comply with a subpoena.

The move would escalate what is shaping up to be a major legal battle between the select committee and the former president over access to crucial witnesses and documents that could shed light on what precipitated the riot, when a pro-Trump mob stormed the Capitol and disrupted Congress’s formal count of the votes that confirmed President Biden’s election.

It came after Mr. Bannon informed the panel that he would defy a subpoena in accordance with a directive from Mr. Trump, who has told former aides and advisers that they should not cooperate with the inquiry because he is claiming executive privilege, which can shield White House deliberations or documents involving the president from disclosure.

“Mr. Bannon has declined to cooperate with the select committee and is instead hiding behind the former president’s insufficient, blanket and vague statements regarding privileges he has purported to invoke,” Representative Bennie Thompson, Democrat of Mississippi and the chairman of the committee, said in a statement. “We reject his position entirely. The select committee will not tolerate defiance of our subpoenas, so we must move forward with proceedings to refer Mr. Bannon for criminal contempt.”

Under federal law, any person summoned as a congressional witness who refuses to comply can face a misdemeanor charge that carries a fine of $100 to $100,000 and a jail sentence of one month to one year.

The committee, which is controlled by Democrats, is expected to agree to pursue such penalties on Tuesday. That would send the contempt citation to the full House, where Democrats almost certainly have the votes to approve it. The matter would then be sent to the Justice Department with a recommendation that officials pursue a legal case against Mr. Bannon.

The cumbersome procedure reflects a challenging reality that Democrats are grappling with as they work to push forward in the inquiry. Congress is a legislative body, not a law enforcement entity, and its ability to compel cooperation and punish wrongdoing is inherently limited. Its investigative tools are only as powerful as the courts decide, and the process of waging legal fights to secure crucial information and witnesses is likely to be a prolonged one.

Robert J. Costello, a lawyer for Mr. Bannon, said in a letter to the committee on Wednesday that his client would not produce documents or testimony “until such time as you reach an agreement with President Trump” on claims of executive privilege “or receive a court ruling.”

The Biden administration has declined to extend the privilege to Mr. Trump, but the question could end up in the courts. And in the case of Mr. Bannon, who has not been an executive branch official since he left the White House in 2017, the claim is particularly tenuous, as it relates to conversations or documents concerning the Jan. 6 attack.

In its first batch of subpoenas, the select committee ordered four former Trump administration officials — Mr. Bannon; Mark Meadows, the White House chief of staff; Dan Scavino Jr., a deputy chief of staff; and Kash Patel, a Pentagon chief of staff — to sit for depositions this week and provide documents and other materials relevant to its investigation.

The committee has said Mr. Meadows and Mr. Patel were communicating with the panel. A source with knowledge of the committee’s negotiations said that lawmakers were likely to grant the two men a delay before testifying. Mr. Scavino was served with his subpoena last week.

On Wednesday, the committee issued a subpoena to Jeffrey Clark, a former Justice Department official who was involved in Mr. Trump’s efforts to overturn the 2020 presidential election. The committee’s action came the same day it heard lengthy closed-door testimony from Jeffrey A. Rosen, the former acting attorney general, who has testified publicly and privately about the final days of the Trump administration, when the former president was pressing top officials to use the Justice Department to advance false claims of fraud and invalidate the election outcome.

In private testimony before the Senate Judiciary Committee, Mr. Rosen said that Mr. Clark had told him that Mr. Trump was getting ready to fire Mr. Rosen and endorse Mr. Clark’s strategy of pursuing conspiracy theories about the hacking of voting booths and election fraud.

“Well, I don’t get to be fired by someone who works for me,” Mr. Rosen said he told Mr. Clark.