In a warning shot on Monday, Speaker Nancy Pelosi informed colleagues that she intended to schedule a full House vote “soon” to hold Mr. Barr and Commerce Secretary Wilbur Ross in contempt of Congress for defying subpoenas for documents related to the census question. Ms. Pelosi called the census dispute “essential to who we are as a nation” and asserted that the materials in question would “shed light on the real reason the administration added a citizenship question.”

The House Oversight and Reform Committee, which is investigating the Trump administration’s decision to add the question, voted last month to recommend that the two cabinet officials be held in contempt, mostly along party lines, despite protests from the administration that it was working in good faith to meet the requests.

If the House follows through with a contempt vote on the floor — and no date for a vote has yet been set — it would be empowering the Oversight Committee to take Mr. Barr and Mr. Ross to court to ask a judge to enforce their subpoenas. Doing so is an exceedingly rare step and puts a black mark on both officials’ public records.

The House has already threatened to hold Mr. Barr in contempt once over a separate case related to a subpoena for material connected to Robert S. Mueller III’s investigation as special counsel. But in the end, lawmakers struck a deal with the attorney general and voted on a resolution that merely authorized them to go to court to enforce the subpoena rather than formally accusing Mr. Barr of being in criminal contempt.

House Democrats intend to go further this time, formally accusing both officials of criminal defiance of their summons if the administration does not relent beforehand, according to a senior Democratic aide familiar with the plans. Still, the practical outcome could be the same since the Justice Department would almost certainly refuse to bring a criminal case against the men.

Kerri Kupec, a Justice Department spokeswoman, declined to comment on the contempt issue.

The conversations between Mr. Barr and Mr. Trump came amid a series of abrupt reversals on the issue. After the Supreme Court delayed the administration’s effort to add the citizenship question, ruling that its rationale was “contrived,” Mr. Ross and Justice Department lawyers declared the issue all but dead last week in the near term.

Mr. Ross said that the Census Bureau, which the Commerce Department oversees, would focus on conducting “a complete and accurate census” and had begun to print forms that did not include the citizenship question. Justice Department lawyers, who had argued that they faced a strict June 30 cutoff for printing the census forms, also concluded as that deadline passed that the question would have to wait for the next census in another decade.