Iran and groups allied with it in Iraq and other countries have been holding events to honour Soleimani, the commander of the Quds Force, the overseas arm of the elite Revolutionary Guards.
He was killed in Iraq in a US drone strike on January 3, 2020.
"If Trump and (former secretary of state Mike) Pompeo are not tried in a fair court for the criminal act of assassinating General Soleimani, Muslims will take our martyr's revenge," Raisi said in a speech on Monday.
"The aggressor, murderer and main culprit - the then president of the United States - must be tried and judged under the (Islamic) law of retribution, and God's ruling must be carried out against him," Raisi added.
Under Iran's Islamic laws, a convicted murderer can be executed unless the family of the victim agree to take "blood money" through a reconciliation.
Iranian judicial officials have communicated with authorities in nine countries after identifying 127 suspects in the case, including 74 US citizens, Prosecutor-General Mohammad Jafar Montazeri told state television.
"The criminal former president (Trump) is at the top of the list," he said.
On Sunday, Iran urged the United Nations Security Council in a letter to hold the United States and Israel - which officials in Tehran say was also involved in the killing - to account, Iranian media said.
Days after the assassination, the US told the United Nations that the killing was self-defence.
The then US attorney general William Barr said Trump clearly had the authority to kill Soleimani and the general was a "legitimate military target".
Hundreds of supporters of Iran-backed militia groups gathered on Sunday at Baghdad international airport to mark the anniversary of Soleimani's death and to chant anti-US slogans.
