"The Iranian regime faces a choice", Mr. Trump said in a statement.
Mr Trump warned that those who do not wind down their economic ties to Iran "risk severe consequences".
British Minister of State for the Middle East Alistair Burt said Europe can protect its companies from new U.S. sanctions which were reinstated on Tuesday, as the European Union activated a blocking statute to insulate firms against any fallout.
"Frankly it is not realistic and I don't think the Iranians are biting because it is pretty clear that this administration is not interested in an agreement with Iran, it's interested in a surrender from Iran".
Hailing the law, Mr Burt said it would protect European businesses.
But his Iranian counterpart Hassan Rouhani dismissed the idea of talks while crippling sanctions were in force.
The Islamic Republic's economy, hobbled by years of sanctions, has suffered in recent months with the uncertainty over the atomic accord. The country's rial currency has tanked, and the downturn has sparked protests across the nation. "Of course, American bullying and political pressures may cause some disruption, but the fact is that in the current world, America is isolated", Mr. Zarif told reporters, according to the semi-official ISNA agency. Renault, Total, Peugeot, General Electric, Boeing, Hyundai, Mazda, IndusInd Bank and other major companies have either left Iran or asked their investors to do so before the return of United States sanctions. Another added, "The Iranian people are seeking the same thing".
National security adviser John Bolton said Iran's leadership is on "very shaky ground", but he insisted economic pressure from the Trump administration is not an attempt at "regime change". It the statement, Daimler states that as for now, neither its car-related nor truck-related activities in Iran had been resumed.
"If the ayatollahs want to get out from under the squeeze, they should come and sit down".
He said that "there's no evidence today of a change in their behavior", and in the meantime "we're going to enforce the sanctions". "That's the ask", said Pompeo on Sunday ahead of the re-imposition.
There is also mounting pressure at home, where U.S. hostility has helped fuel long-running discontent over high prices, unemployment, water shortages and the lack of political reform.
The European Union has sought to salvage the nuclear deal and provide legal cover for its companies to continue operating in Iran.
Prime Minister Benjamin Netanyahu said the sanctions symbolize "the determination to block Iran's regional aggression as well as its continuous plans to arm itself with nuclear weapons".
He added, "To this day, Iran threatens the United States and our allies, undermines the worldwide financial system, and supports terrorism and militant proxies around the world".