The order would allow Trump to sanction foreigners who interfere in the midterm elections to be held in less than two months.
The order comes as intelligence agencies, the military and law enforcement prepare to defend the November congressional elections from foreign attacks.
As The Washington Post first reported in August, the order appears to be an effort to stave off bipartisan legislation that would mandate tough federal action.
Press secretary Sarah Sanders later clarified Trump meant to say he had no reason to believe it "wouldn't" be Russian Federation, using the wrong word - but at that point, the damage of the remarks and the skepticism he expressed had already led to bipartisan outcry, particularly as intelligence officials warn that Russian Federation may continue to attempt to attack United States elections as the midterms approach in November.
Mr Trump has been criticised for his response to alleged Russian meddling.
Rogers, who spoke Tuesday night at the Hayden Center at George Mason University in Virginia, also said earlier media reports claiming Trump had asked him to publicly deny any collusion between Moscow and Trump's campaign were inaccurate.
Trump has repeatedly dismissed the investigation as a "witch hunt".
That this announcement came first as a leak to United States news outlets and then as an announcement from National Security Advisor John Bolton underlines how fraught a subject it is for this particular White House.
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He said Secretary Kirstjen Nielsen "is confident in the leadership at FEMA and their proven disaster management ability", he said. Representatives for the inspector general and the White House did not immediately respond to requests for comment.
"Today's announcement by the Administration recognizes the threat, but does not go far enough to address it", they argued.
Trump signed executive order that initiates actions, including sanctions, against foreign entities who meddle or try to interfere in polls.
Both men gave a briefing on Wednesday outlining details of the executive order, which creates a framework to act should election meddling be identified. The order would direct intelligence agencies to assess whether any people or entities interfered.
The order declares a national emergency, Bolton said, and requires DNI to make regular assessments of activities targeting the USA electoral process, reporting findings to the Department of Justice and Department of Homeland Security. Intelligence agencies would have 45 days to make an assessment.
Then the U.S. attorney general and Department of Homeland Security will have a further 45 days to decide whether to go forward with sanctions. The executive order, called "Imposing Certain Sanctions in the Event of Foreign Interference in a United States Election", establishes a mechanism to impose sanctions on any foreign individual or company found interfering. He said he was in talks with lawmakers about possible legislation.
But critics said that Trump's move is not tough enough, and leaves him too much discretion, after the broad assault on the U.S. electoral process by Moscow two years ago. Its findings would then be turned over to U.S. Attorney General Jeff Sessions and Department of Homeland Security, which would have another 45 days to determine what action to take. Coats also highlighted potential meddling from China, Iran and North Korea.
"This is not a single solution, but it makes a clear statement by the president that this sort of activity will not be tolerated and will be punished", the official said.
Congress passed a financial sanctions bill against the Kremlin over a year ago and since January 2017, over 200 individuals and companies with connections to Russian Federation have been subjected to its penalties.