One 25-year-old from Mbare, a populous township, said he was targeted for being a political agent for the opposition Movement for Democratic Change (MDC).
"I need to go out of this country, I don't feel safe," he said.
One woman said she and her one-year-old daughter had been left homeless by suspected government agents who burned down her home in a rural district east of the capital.
"About a month ago some people came at night saying my family were MDC supporters and burned our house," she said.
"We managed to escape in different directions and sleep in the bush but we lost everything. Up to now I don't know where my husband is.
"I need food and shelter. That is my appeal to anyone who can help."
Our correspondent said the people were desperate, feared for their lives and could not go home as homes had been destroyed.
She reported that embassy officials were trying to help the people but they said the Zimbabwean government should be looking after them.
The US officials also blamed the government for not ending the violence that has displaced the people.
War veterans, who support President Robert Mugabe's ruling Zanu-PF party, also arrived at the embassy on Thursday.
|The MDC supporters say they are scared
for their lives [AFP]
They said that the MDC's claims and those of its supporters were false.
The veterans have called for MDC supporters' removal from outside the embassy.
More than 300 opposition supporters sought refuge at the South African embassy in Harare last week.
They were subsequently taken to a camp outside the capital by government officials.
Our correspondent said concerns had surfaced as to what would happen to that group as there were reports that some of the people looking after them were the same people who had been burning their homes or beating them up.
Those seeking refuge in the US embassy are likely to receive a similar fate, she added.
The MDC won parliamentary elections marred by violence on March 29.
It subsequently pulled out of a presidential run-off last week, saying 90 of its supporters had been killed in a government campaign of violence and intimidation.
The US administration has led criticism of Robert Mugabe, the Zimbabwean president, calling the run-off election a farce.
On Thursday the US lodged a protest with the UN over the run-off vote and pushed for a travel ban and asset freeze on Mugabe and over a dozen of his close associates for their roles "in abetting the state-sponsored violence against the opposition, repressing human rights or undermining democracy".
Zalmay Khalilzad, the US ambassador to the UN, introduced a resolution condemning Mugabe's government and containing targeted sanctions including an arms embargo for the security council to vote on next week.
The US also wants the UN to appoint "an individual of international standing and expertise ... who would support the negotiation process between the political parties in Zimbabwe".
Diplomatic sources say Kofi Annan, the former UN secretary-general, is being considered for the task.