A threat by House conservatives to impeach Deputy Attorney General Rod J. Rosenstein stands to linger for at least another month, carrying implications for the Russia investigation, the future of the House Republican leadership and the midterm elections.

Members of the hard-line House Freedom Caucus decided not to force a vote on their 12-page impeachment resolution Thursday, the last day lawmakers were scheduled to meet in Washington before September. But they delivered a new ultimatum for the production of investigative documents from the Justice Department and opened a rift among GOP leaders over how to deal with their demands.

House Speaker Paul D. Ryan (R-Wis.) said flatly Thursday that he opposed the effort to impeach Rosenstein and that he believed Justice Department officials were working in good faith to comply with the congressional oversight requests.

“I don’t think this rises to ‘high crimes and misdemeanors,’ ” he told reporters, raising the specter that an impeachment trial could tie the Senate in knots, delaying top-priority conservative agenda items such as the confirmation of Judge Brett M. Kavanaugh to the Supreme Court.

But two other leaders angling to succeed Ryan after he retires next year were more sympathetic, offering support to the effort to unseat a top law enforcement official of their own party as they continue wooing support from conservatives.

House Majority Whip Steve Scalise (R-La.) voiced support for the impeachment measure, calling it “leverage” to get the Justice Department to provide Congress with more documents related to the origins of the Russia probe, led by special counsel Robert S. Mueller III, that Rosenstein now oversees.

Scalise, who is the third-ranking Republican in the House and hopes to move up the ladder, said he would vote for the resolution if it reached the floor.

“This is another tool to get Justice to comply with our subpoenas and our demands for documents that the American people deserve to get,” he said. “They need to start complying. Obviously they have given us a number of things, but they’ve still held back some of the documents we need to get as part of our oversight.”

House Majority Leader Kevin McCarthy (R-Calif.) suggested slowing down the process, saying that the Justice Department should comply with House requests but that the resolution should be dealt with in a committee first before being considered in the full House.

“What everybody’s trying to do is get the information,” McCarthy said. “I think what would make this resolution go away is to supply the information to the House.”

McCarthy and Scalise faced a new challenge from the right Thursday as Rep. Jim Jordan (R-Ohio), the hard-charging conservative who helped spearhead the impeachment resolution, announced his candidacy for speaker.

The Justice Department insists that it has cooperated with the demands from Congress. Officials have said that they have provided the vast majority of information sought in subpoenas from two key House committees and are nearly done with providing all the outstanding information requested in those subpoenas.

The agreement with conservatives — forged in hours of private conversations that culminated in a massive morning huddle on the House floor — essentially puts off the showdown until lawmakers return from their five-week recess. According to members familiar with the agreement, House leaders will present the Justice Department with a final list of requested documents, and failure to hand them over by September will result in contempt proceedings.

Rep. Mark Meadows (R-N.C.), who filed the impeachment measure with Jordan and nine other members of the Freedom Caucus, said conservatives decided it was “most prudent” to “give the FBI and DOJ one more benefit of the doubt” before taking action against Rosenstein. They have accused the No. 2 federal law enforcement official of withholding documents and being insufficiently transparent in his handling of the Russia probe, and they did not rule out pursuing impeachment at a later date.

“All options are still on the table and remain on the table,” Meadows said.

Many Republicans questioned the wisdom of the impeachment effort, including some reliable conservatives. A showdown vote could loom large just weeks before the November elections.

Rep. Mark Walker (R-N.C.), chairman of the Republican Study Committee, said pursuing impeachment was “putting the cart before the horse.”

“If we have a problem overall with how the DOJ is operating, we should be putting pressure on the attorney general and not be going after one of the deputies,” he said. “We kind of got a little bit catawampus here.”

Rep. Ken Buck (R-Colo.), a Freedom Caucus member and former prosecutor who has been involved in discussions with the Justice Department on the requests, said he was pleased with the deal reached Thursday. “It is far better than the moron attempt to impeach the deputy attorney general,” he said.

“Reckless publicity stunt,” Rep. Carlos Curbelo (R-Fla.) tweeted Wednesday night. “No different from Dems who filed articles of impeachment against the President some months ago. What a sad, pathetic game of ‘how low can you go?’ ”

Meanwhile, Attorney General Jeff Sessions expressed confidence in Rosenstein when asked Thursday about the impeachment effort during a news conference in Boston.

“My deputy, Rod Rosenstein, is highly capable,” Sessions said. “I have the highest confidence in him.”

Democrats have said that House Republicans’ clashes with Rosenstein are little more than a pretext to weaken Mueller’s efforts.

Rep. Adam B. Schiff (D-Calif.), the top Democrat on the House Intelligence Committee, said that the push to get more documents “really has nothing to do with oversight.”

“This has everything to do . . . with getting the documents to the president’s allies in Congress so [Trump attorney Rudolph W.] Giuliani can get his hands on them,” Schiff said, adding that Republicans were also trying to give President Trump “a pretext” to fire Rosenstein “so he can replace him with someone who will simply do his bidding.”

Meadows disputed that the House conservatives, who are among Trump’s most loyal backers on Capitol Hill, were pursuing impeachment to undermine the Mueller probe.

“The Mueller investigation is an independent special counsel,” he said. “I can tell you, based on the documents I’ve seen, I can tell you Rod Rosenstein is not supervising that to any great extent. It is driven by Bob Mueller.”

Rosenstein’s removal, however, would give Trump the opportunity to nominate a new No. 2 at the Justice Department who might be more willing to rein in Mueller. Sessions has recused himself from the probe, citing his influential role in the 2016 Trump campaign.

Karoun Demirjian contributed to this report.