He was killed in a US airstrike in Baghdad, Iraq, on Friday, causing a huge surge in tensions in the Middle East.
The US president had warned that the US could retaliate “perhaps in a disproportionate manner" if Iran strikes a US citizen or target.
Mr Trump also claimed Iranian cultural sites are fair game for the US military, dismissing arguments that doing so would constitute a war crime under international law.
On Monday, the prime minister’s official spokesman warned that an attack on a cultural target is prohibited under the Hague Convention, of which the US is a signatory.
“There are international conventions in place that prevent the destruction of cultural heritage,” Mr Johnson’s spokesman said.
However, Downing Street insisted that Britain's security partnership with the US remains "very close”, despite Mr Trump not informing the UK of its plans to assassinate Soleimani.
Asked if Mr Johnson was convinced the US drone strike was legal, his spokesman said: "States have a right to take action such as this in self-defence and the US have been clear that Soleimani was plotting imminent attacks on American diplomats and military personnel."
Number 10 urged the Iraqi government to allow foreign troops to remain in the country to fight against the threat posed by Isis.
On Sunday evening, Mr Johnson, along with his French and German counterparts Emmanuel Macron and Angela Merkel, called for a “de-escalation” in the Middle East tensions.
Earlier on Sunday, Mr Johnson said Soleimani had been "a threat to all our interests" and said: “We will not lament his death”.
But Iran has vowed “severe revenge” for the death of Soleimani, 62, the head of the elite Quds Force, pulling back from the 2015 nuclear accord.
Officials in Iran said “millions” of people had taken to the streets of the capital Tehran for the general’s funeral on Monday.
People packed the streets of the city and mourners were heard shouting, “Death to America”.
In a televised address, the general’s daughter Zeinab Soleimani warned the US it faced a “dark day” for her father’s killing.
She said: "Crazy Trump, don't think that everything is over with my father's martyrdom.
"The families of the American soldiers in western Asia will spend their days waiting for the death of their children.”
Iran's supreme leader Ayatollah Ali Khamenei prayed over the caskets of Soleimani and others slain in the US attack.
Khamenei, who had a close relationship with Soleimani, wept at one point during the traditional Muslim prayers for the dead.
Soleimani's successor Esmail Ghaani stood near Khamenei's side, as did Iranian president Hassan Rouhani.
Amid fears of an all-out war, Iraq's parliament has passed a non-legally binding bill calling for the expulsion of all foreign forces.
About 400 UK troops are stationed in Iraq in the fight against Isis, while the US has 5,200, prompting fears of a withdrawal that could cripple the battle against the terror group.