Rosen told Trump in the extraordinary 27 December 2020 discussion that the department had looked into his claims of voter and ballot fraud and found no evidence.
"Understand that the DOJ can't + won't snap its fingers + change the outcome of the election, doesn't work that way," Rosen told Trump, according to the notes taken by deputy attorney general Richard Donoghue.
But Trump was desperate to reverse his November election defeat by now-president Joe Biden.
"We have an obligation to tell people that this was an illegal, corrupt election," he said.
The record of Trump's pressure on the department, just a week before his supporters stormed the US Capitol in an attempt to stop the certification of the election results, was released Friday by the House Committee on Oversight, which is investigating Trump's unprecedented efforts to overturn the vote.
"These handwritten notes show that president Trump directly instructed our nation's top law enforcement agency to take steps to overturn a free and fair election in the final days of his presidency," Carolyn Maloney, the committee chair, said in a statement.
The pressure on Rosen came days after Trump had fired attorney general Bill Barr for refusing to accept the real estate mogul's claims of election fraud.
Barr said in a recent interview that he told Trump at the beginning of December that the fraud claims were "bullshit."
The Washington Post reported Thursday that after Rosen replaced Barr, Trump called him "nearly every day" to press the fraud claims.
Trump mused to Rosen that he could be replaced with a lower-level Justice Department attorney, Jeffrey Clark, who, according to later reports, actually helped the president devise a plan to force Rosen out.
Clark, according to reports, supported Trump's claims of election fraud, though no evidence has ever surfaced of significant wrongdoing in any of the states or districts that delivered the election to Biden.
The House committee is citing Donoghue's notes and other records to call him, Clark, Trump's White House chief of staff Mark Meadows and other top officials to testify about Trump's campaign to reverse the election.
Earlier this week, the Justice Department said it would not seek to assert White House executive privilege to block their testimony.
Meanwhile, the watchdog group Citizens for Responsibility and Ethics in Washington (CREW) filed a new lawsuit Thursday calling for the Justice Department to investigate Trump and Meadows for violating federal civil rights law.
The two, CREW alleged, illegally attempted "to weaponize the Department of Justice as part of their larger campaign to overturn the results of the 2020 presidential election."
"These alarming and illegal acts were part of a broader conspiracy to deprive American citizens of their right to vote and to have their votes counted," said CREW president Noah Bookbinder.