The release this week of new information from the estate of the convicted sex offender Jeffrey Epstein, including a suggestive note to him apparently signed by Donald J. Trump, has not quieted the clamor on Capitol Hill for full transparency from the Justice Department about Mr. Epstein’s case.

Despite staunch opposition from the White House and Republican leaders, a bipartisan resolution directing the Justice Department to release all of its investigative files on Mr. Epstein is still on track.

Its proponents, Representatives Thomas Massie, Republican of Kentucky, and Ro Khanna, Democrat of California, appear poised within weeks to draw enough backers to force action on the House floor, provided Democrats win special elections this month in districts where they are heavily favored.

But while their effort is expected to continue being a political headache for Republican leaders and the White House, there is still no guarantee it will receive a vote, and little chance that the House would compel the Justice Department to comply even if it passed.

Here’s how it works.

A House majority can force action, even if leaders object.

Speaker Mike Johnson and Trump administration officials had hoped they could thwart Mr. Massie and Mr. Khanna’s effort to force their measure to the floor.

But Mr. Massie, a frequent Trump antagonist, is on the brink of drawing enough support from the House to move forward. Last week, he introduced what is known as a discharge petition, a maneuver that allows lawmakers to circumvent party leaders and bring a bill to the floor if 218 lawmakers demand it.