Although Rosenstein didn't mention any specific media reports, it appears it could be in response to a Washington Post story published Wednesday, alleging that the special counsel in the Russian Federation probe is investigating President Trump for potential obstruction of justice, related to his firing of then-FBI Director James Comey.
It wasn't clear whether the president's comment confirming he was under investigation was based on direct knowledge or media reports this week that Mueller is examining whether the president obstructed justice by firing Comey last month.
But some outlets pointed out the statement was released after both The Washington Post and The New York Times reported Special Counsel Robert Mueller is investigating President Donald Trump for obstruction of justice.
Yesterday, Trump referred to the possible obstruction of justice investigation as the "single greatest witch hunt in American political history".
Sessions said only the president has to power to invoke executive privilege but explained that neither he nor Trump had chose to do so.
Richard Cullen is a former Virginia attorney general and a former United States attorney for the eastern district of Virginia.
The Democratic National Committee urged June 16 for Rosenstein to also recuse himself, arguing that Trump has shown little hesitance to fire those he sees as opponents.
Mr Comey said he was "sure" Mr Mueller was looking at whether Mr Trump had obstructed justice, but added that to his knowledge, the president had not tried to impede the Russian Federation inquiry.
Trump has denied any involvement with the Russians during the campaign and has called media reports of collusion a "hoax" and a "phony story".
The order also demands that records be kept on any foreign travel by members of Trump's transition team and presidential campaign.
Trump's young presidency has been battered by allegations - under investigation both by Congress and the FBI - that Russian Federation interfered to sway the 2016 election in his favor, in possible collusion with Trump's campaign team.