Relatives of 26-year-old Erfan Soltani, who was detained last week, have told BBC Persian that he is due to be executed on Wednesday. BBC reported.
A representative from the Hengaw Organization for Human Rights also told the BBC that they had "never witnessed a case move so quickly".
Speaking to the BBC's US partner CBS News, Trump said of potential executions: "If they hang them, you're going to see some things... We will take very strong action if they do such a thing."
Soltani's relative told BBC Persian that an Iranian court had issued a death sentence "in an extremely rapid process, within just two days".
Awyar Shekhi, a representative of Hengaw, said that the case demonstrated that the Iranian government is "using every tactic they know to suppress people and spread fear".
An Iranian official told Reuters that 2,000 people had been killed, but that "terrorists" were to blame.
Trump earlier said he planned to attend a meeting at the White House on Tuesday night to discuss the situation in Iran, pledging to get "accurate numbers" on the death toll from the protests.
The US-based Human Rights Activists News Agency (HRANA) said that it had so far confirmed the killing of 2,403 protesters, as well as 12 children, despite an internet blackout. Nearly 150 people affiliated with the government have also been killed, the group said.
"The killing looks like it's significant, but we don't know yet for certain," Trump told reporters while returning to the White House.
Once he has the numbers, he said, "we'll act accordingly."
