Rosenstein was also the author of a letter in May to Trump criticizing the performance of FBI Director James Comey.
When President Barack Obama entered office, Rosenstein was asked to stay on and served for all eight years of the president's term.
Aides have counseled the president to stay off Twitter and focus on other aspects of his job.
There has been many twists and turns during the 2016 Presidential election, take a look at some of the key moments which won and lost the race for the White House.
Rod Rosenstein probably never expected to be in the spotlight so often while serving in President Donald Trump's administration. Trump himself took aim Friday at the senior Justice Department official responsible for appointing Mueller, accusing him on Twitter of leading a "Witch Hunt". They are Paul Manafort, Trump's former campaign manager; Rick Gates, Manafort's business partner; Carter Page, a former foreign policy adviser; Flynn; and Roger J. Stone Jr., an informal adviser to Mr. Trump.
Mark Corallo, a spokesman for Trump's personal lawyer, had responded Wednesday evening to the Post report by saying: "The FBI leak of information regarding the president is outrageous, inexcusable and illegal".
In a statement late Thursday, Rosenstein cautioned Americans against "accepting as true any stories attributed to anonymous 'officials, ' particularly when they do not identify the country - let alone the branch or agency of government - with which the alleged sources are supposedly affiliated".
The US Justice Department has assigned Mueller to investigate the Russian meddling allegations. Flynn was forced to resign for misleading administration officials about his contacts with Russia's USA ambassador. The American leader also slammed Comey for disclosing his memos to the press. It's possible that Rosenstein ordered the obstruction probe before Mueller's appointment in May.
Once Comey was sacked, the Department of Justice appointed Mueller as special counsel to act as an independent investigator.
Reporters who shouted questions for clarification after Trump stepped off Marine One on his return to the White House on Friday from a Miami trip got no answers.
He is the department's lead official on the issue after Attorney General Jeff Sessions recused himself from any issues linked to the Russian Federation probe.
That suggests Mueller is examining the president's own conduct, which may include whether Trump tried to obstruct justice.
After rising through the ranks, Rosenstein was unanimously confirmed as United States attorney for the District of Maryland in 2005 under then-President George W. Bush.
Mueller's team has also invited other intelligence officials to give their testimonies so the special counsel could rule if there is a solid case against Trump. An official of Trump's transition confirmed the lawyer's internal order, which was sent Thursday. The presumption, however, is that Mr. Comey's dismissal was due to official inquiries he was making about Trump's inner circle, namely, former National Security Adviser Michael Flynn, and its ties with Russian Federation.
Former transition team members "have a duty to preserve any physical and electronic records that may be related in any way to the subject matter of the pending investigations", the memo warns.
A few days later, Mr Trump individually asked Mr Coats and Mr Rogers to issue public statements to the effect that there was no evidence of co-ordination between his campaign and Russian Federation.
Meanwhile, another lawyer, Washington veteran John Dowd, has joined the Trump legal team.