The United States will keep its embassy and other American government offices in South Africa closed on Wednesday after an undisclosed security threat, the embassy said on its website on Tuesday.
U.S. embassy spokeswoman Sharon Hudson-Dean had earlier said the assumption was that the U.S. offices would reopen on Wednesday.
South African Police Commissioner Bheki Cele did not give details of the threat and said he could not rule out the possibility of a hoax while investigations continued.
It is under control, he told reporters in Cape Town.
The threat might be genuine itself or the threat might not exist, it might be a hoax. If our investigation puts some concrete steps that we must follow, there will be arrests, he said, adding police were pursuing some people.
South Africa, due to host the soccer World Cup next year, is not itself seen as a target for attacks.
Somali rebels have vowed to avenge the killing last week of one of the continent's most wanted al Qaeda suspects in a raid by U.S. commandos, but there has been no previous link between Somali insurgents and South Africa.
Bombings of U.S. embassies in Kenya and Tanzania in 1998 killed 224 people and wounded thousands.