White House National Security Adviser Robert O'Brien also defended the strike, saying the Iranian regime is "having a very bad week" and that the US would continue a "maximum pressure campaign" against the regime.
He also said the president has shown "incredible restraint" in the face of regular provocation from Iran and has also been "modest in his dealings" with other countries.
But O'Brien did not confirm Trump's claim that the White House had received intelligence that Soleimani, the head of Iran's elite Quds Force, was planning "imminent" attacks against four US embassies.
"What the President said is consistent with what we've been saying. We had very strong intelligence that they were looking to kill and maim Americans in American facilities in the region," O'Brien said on Fox News.
Even with America's "exquisite intelligence" it is difficult "to know exactly what the targets are," O'Brien said. He added that it was fair to anticipate a future Iranian attack "would have hit embassies in at least four countries."
Pressed on why the White House has not revealed more details on the alleged threat they say precipitated the Iranian strike, O'Brien said, "I would love to release the intelligence," but "those same streams and channels" are important to protecting Americans.
Top Democrats have also pushed back on Esper's claim that the Gang of Eight - the bipartisan Congressional group that traditionally is read in on classified intelligence and military maters - was given information on the threat to attack the embassy in Baghdad.
Representative Adam Schiff, D, a member of the Gang of Eight, contradicted Esper's assertion about the briefing to Congress, saying it lacked "specificity" about a potential embassy threat. Schiff said he and several members of the Gang of Eight were dissatisfied with the evidence laid out as a basis for the strike.
Trump and Esper are "fudging" the details, Schiff added, and "overstating and exaggerating what the intelligence shows." When it comes to information that could lead to a potential war in Iran, he said, "that's a dangerous thing to do."
Trump's claim about threats against embassies was also not part of a Senate briefing last week, Senator Mike Lee, R, said on CNN.
"That was news to me," he said. "It certainly wasn't something I recall being raised in the classified briefing."
Lee also savaged the Trump Administration for failing to sufficiently justify the strike. He earlier called the briefing the "worst" he's received in nine years in the Senate".
