Pompeo did not name the countries to be exempted, except to say that the European Union was not among them.
U.S. sanctions targeting Iran's oil sales and banking sectors are being restored on Monday as part of a wider effort by Mr Trump to force Tehran to halt its nuclear and ballistic missile programmes outright.
General Qassem Soleimani, commander of the elite Iranian Revolutionary Guards' Quds Force that oversees its operations overseas, responded to Trump's Game of Thrones-inspired tweet on Friday that warned, "Sanctions are Coming".
Iran is heavily dependent on its exports of oil, and renewed sanctions, if effective, would hit the economy hard. South Korea, Japan and India are expected to get a reprieve, however, and possibly China.
"Iran has around 30 vessels in the Gulf area, so the past 10 days have been very tricky, but it hasn't slowed us down".
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"In the current geopolitically challenging scenario, India has managed to convince worldwide leaders", Pradhan told reporters in New Delhi.
Oil prices fell on Friday for a weekly loss of over 6 percent, as investors anxious about crude oversupply after the USA waivers.
Oil prices fell on Friday for a weekly loss of over 6 percent on worries about crude oversupply due to the USA waivers.
Treasury Secretary Steve Mnuchin defended the decision to allow some Iranian banks to remain connected to SWIFT, saying that the Belgium-based firm had been warned that it will face penalties if sanctioned institutions are permitted to use it.
These economic sanctions are just a part of the United States government's total effort to change the behaviour of the Iranian regime, he said.
As a result of the latest sanctions, he said the USA expects to have reduced Iranian crude oil exports by more than 1 million barrels even before these sanctions go into effect. "The other six will import at greatly reduced levels", Pompeo said.
However, Mr Trump said the deal left Iran free to destabilise the region by funding militias and using its influence to further its interests.
"We will ultimately move Iran to zero crude oil".
Demonstrators in Tehran gathered on Sunday outside the former USA diplomatic mission, also known as the "Den of Espionage", chanting slogans against the United States.
The US had insisted on cutting exports to zero but that seems unlikely as it would increase the price of oil, says Scott Lucas, a professor of worldwide politics at Birmingham University. "That's a good thing for American consumers".
The measures will take effect from Monday and cover Iran's shipping, financial and energy sectors.
"America will not be able to carry out any measure against our great and fearless nation".
"I can not even buy rice to feed my children or pay my rent", said elementary school teacher Pejman Sarafnejad, 43, a father of three in Tehran. Announcing that India is one of the companies to get a waiver in the United States sanctions on Iran, Pradhan said that India has been able to get its own way. The Europeans have mounted a drive to save the agreement without the USA, fearing that the new sanctions will drive Iran to pull out and resume all of its nuclear work.