Though Mr. Graham, the Senate Judiciary Committee chairman, said he supported transparency, he asked to amend the resolution to include the appointment of a new special counsel to investigate the F.B.I.’s handling of the Hillary Clinton email case and its surveillance of a former Trump adviser. Mr. Schumer rejected the request as political and beside the point, and the resolution failed.

The Justice Department has given signals in recent weeks that after 22 months, Mr. Mueller is nearing completion of his work. Peter Carr, a spokesman for Mr. Mueller, offered what many viewed as fresh evidence of an imminent conclusion on Thursday when he confirmed that one of the special counsel’s top prosecutors, Andrew Weissmann, will be departing the special counsel’s office “in the near future.”

The resolution — sponsored by Representative Jerrold Nadler of New York, the Judiciary Committee chairman, and a handful of other Democratic committee leaders — “calls for the public release of any report Special Counsel Mueller provides to the attorney general, except to the extent the public disclosure of any portion thereof is expressly prohibited by law.”

Under the regulation that governs special counsels, Mr. Mueller is expected to produce a confidential report on his prosecution decisions to Mr. Barr, who will then review it and produce his own report to Congress. Thus far, Mr. Barr has demurred on just what he will release to Congress and the public, reserving the right to keep some matters secret.

Democrats seized on Mr. Barr’s resistance to making specific promises about the Mueller findings during his confirmation process in the Senate. They have not let the point rest in the weeks since, with prominent Democratic chairmen and other leaders laying out their case for why all of the special counsel’s findings — including underlying evidence — ought to be shared with Congress for review.

House Democrats are prepared to use subpoena power and other tools at their disposal to force the Justice Department to turn over anything Mr. Barr chooses to withhold.

They argue that the Justice Department set a new precedent in the last two years when it granted Republicans, who then controlled the House, extraordinary access to hundreds of thousands of pages of sensitive investigative material related to the F.B.I.’s investigation of Mrs. Clinton’s use of a private server and the Russia investigation itself.