Rosenstein, who is supervising special counsel Robert S. Mueller III's investigation into Russia's interference in the 2016 election, said the Department of Justice was not convinced the memo accurately describes its investigative practices. He said making the document public could set a dangerous precedent, according to a person familiar with the discussion.
While Wray also expressed opposition to the memo's release, Rosenstein did much of the talking, according to a senior U.S. government official. Attorney General Jeff Sessions was not present at the meeting.
In response, Kelly told Rosenstein and Wray that the president was still inclined to release the memo but the White House would go through a review led by the National Security Council and the White House Counsel's Office, a senior administration official said. That review is expected to take at least several days, a senior White House official said.
Trump has not yet seen the memo, according to a White House spokesman.
The West Wing meeting Monday with Kelly, which was originally scheduled to discuss other matters, is the latest sign of the tension between the administration and the Justice Department. That divide has deepened in the charged debate over the memo detailing alleged surveillance abuses by the FBI and the Justice Department, a document championed by Rep. Devin Nunes (R-Calif.), chairman of the House Intelligence Committee.
Rosenstein's urgent push to head off the disclosure of the classified memo came after Kelly privately relayed to Sessions last week that Trump wants to see the document come out, as The Washington Post reported Saturday.
The effort also comes amid questions about Rosenstein's own future. The president has told close advisers recently that the memo could provide him with grounds for either firing or forcing the deputy attorney general to leave, according to one person familiar with his remarks.
The White House, the Department of Justice and the FBI all declined to comment.
The showdown over the memo comes after Trump has lashed out at a number of top Justice Department officials. After firing FBI Director James B. Comey last spring, he has told aides at times that he would like to oust Sessions and Mueller. The president backed off a threat to fire the special counsel last June after White House Counsel Donald F. McGahn threatened to resign.
Trump's interactions with senior law enforcement officials have drawn scrutiny from Mueller's team, which is probing potential obstruction of justice.
White House officials have downplayed the Justice Department's role in the decision whether to release the memo. On Monday, deputy White House press secretary Raj Shah said on CNN that "the Department of Justice doesn't have a role in this process."
The release is increasingly probable after the House Intelligence Committee voted along partisan lines Monday night to release the memo, giving the president a five-day window to decide whether he agrees with the release. Leading Republicans have said it uncovers abuses that should be made public. Democratic lawmakers have said the document is a political tactic to undermine the Mueller investigation.
People familiar with the Intelligence Committee's memo say its main target is the FBI's relationship with Christopher Steele, a British ex-spy who was hired in 2016 by a Washington research firm to examine any connections between Trump and Russian leaders. The work, which was funded by Clinton's presidential campaign and the Democratic National Committee, led to a dossier of allegations against Trump and some of his advisers — allegations that the president has denied.
Within the FBI, some of Steele's work was eventually incorporated into a 2016 application to the Foreign Intelligence Surveillance Court to conduct surveillance on Carter Page, a former Trump campaign adviser, according to people familiar with the matter. Republican lawmakers have suggested that Steele provided bad information to the FBI, leading to a broader probe of Trump associates.
People familiar with the memo said it does not conclusively say whether Steele intentionally passed suspect information to the FBI or simply made a mistake.
Matt Zapotosky contributed to this report.