The attack that reportedly prompted Bolton to seek retaliation involved three mortars fired into Baghdad's sprawling diplomatic quarter, home to the U.S. Embassy.
The mortars landed in an empty lot, and no injuries were reported.
The request to the Pentagon came from U.S. National Security Adviser John Bolton, and it is unknown if Bolton had conversed with the President prior to making the request.
There was also an air strike near a US diplomatic facility in Basra, Iraq.
"It definitely rattled people", a former senior administration official told the paper.
Speaking to an audience of anti-Tehran activists, the longtime Iran hawk said: "The murderous regime and its supporters will face significant consequences if they do not change their behaviour". The Journal reported that it wasn't clear whether Trump knew about, or was provided with, the plans for potential military strikes.
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In one NSC meeting, then deputy national security advisor Mira Ricardel called the attacks in Iraq an "act of war" and called for a decisive U.S. response.
Secretary of Defense James Mattis, along with other members of the military cautioned against retaliating, according to the Times, which wrote, "Mr. Bolton does not want to hear opposing views, these officials said, abhors leaks and wants to control everything that flows to the president".
Garrett Marquis, a spokesman for the National Security Council, told the newspaper that the council "coordinates policy and provides the president with options to anticipate and respond to a variety of threats".
The State Department did not comment on the report. "America will respond swiftly and decisively in defense of American lives", it added.
Mr Netanyahu said: "The accumulation of recent attacks proves that we are determined more than ever to take action against Iran in Syria, and we will strike harder in Syria if needed".
As a think tank scholar and Fox News commentator, Bolton often urged Washington to attack Iran, including in a 2015 New York Times op-ed titled: "To stop Iran's bomb, bomb Iran".