House Republicans, under pressure from Democrats and their own angry constituents, agreed on Thursday night to lay the groundwork for a potential vote calling on the Justice Department to release material from its investigation of the disgraced financier Jeffrey Epstein, breaking with President Trump’s demand that supporters accept his administration’s handling of the case and move on.

The measure, a nonbinding resolution, has not been scheduled for a vote. It is unclear whether Republicans will ever bring it up, or whether it could muster the support to pass in the face of a deep G.O.P. divide over the Trump administration’s handling of the case of Mr. Epstein, who died by suicide in federal custody while awaiting trial on sex trafficking charges.

But Republicans on the powerful House Rules Committee voted to advance it after Democrats threatened to force additional votes on the Epstein disclosures, which could have put G.O.P. lawmakers in a tough spot with constituents who are clamoring for more information. The dispute was holding up final passage of legislation to claw back $9 billion in congressionally approved spending, which had been scheduled for a vote on Thursday evening.

Even as the panel approved the Epstein measure, Mr. Trump, who has faced days of backlash from loyal supporters over the Justice Department’s decision to close its investigation without releasing additional information, said that he would authorize Attorney General Pam Bondi to release testimony from grand jury proceedings in the case.

“The effect of this resolution is much stronger than any of you had expected,” said Representative Virginia Foxx of North Carolina, the Republican chairwoman of the Rules Committee, apparently suggesting that it had swayed Mr. Trump to authorize Ms. Bondi to release the testimony.

But the approval of the resolution by the Rules Committee, a panel dominated by Republicans and generally under the tight control of the speaker, reflected that widening tension between Mr. Trump, his base and Congress over the case of Mr. Epstein had reached a breaking point.