The request was made shortly after Rosenstein took office last spring, when Attorney General Jeff Sessions recused himself from the department's Russian Federation investigation, said three people familiar with it, according to the New York Times.
Rosenstein, who became the No. 2 official at the Justice Department in April and now oversees the investigation into Russia's influence in the 2016 election, approved an application past year to extend surveillance of Trump ally Carter Page, according to the report.
The FBI had been tracking Page, who was previously based in Moscow, since 2013, but was never charged with any wrongdoing.
The secret memo - which has pushed Republican members of the House to demand its release - reportedly claims that officials did not tell an intelligence court judge that they were partially relying on information from the Christopher Steele dossier in order to obtain the warrant.
A trip Page took to Russian Federation in July 2016, while working on Trump's campaign as a foreign policy adviser, caught the attention of the bureau, and law enforcement began conducting surveillance on him in the fall of 2016, shortly after he left the campaign, reported the NYT.
The committee could hold a vote on the memo as early as Monday.
Because Rosenstein extended the surveillance, the memo alleges, it is possible he did so on the basis of unverified information - and tarnished the name of a high-profile Trump associate in the process. Americans deserve the truth.
The White House has backed releasing the memo public. The hashtag #ReleaseTheMemo has gone viral.
But Democrats who have seen the four-page memo - written by House Republicans - say they carefully selected information that is meant to discredit the investigation into Russian involvement in the election and any collusion on the part of the Trump campaign.
No information has publicly emerged, however, that the DOJ or Federal Bureau of Investigation did anything improper.
The memo, which has been the source of bipartisan squabbling over the past week, is said to allege that the Justice Department and Federal Bureau of Investigation abused the FISA process to obtain the order against Page, a 46-year-old energy consultant who joined the Trump campaign in March 2016.
The development may signal that Rosenstein could face renewed criticism from within his own party, including from the White House.
"To the extent that the House, I think, has advocated that it's publicly released, I think the president is receptive to that", he said.