The President withdrew America from the historic Iran agreement last week, but other nations, including the UK, France, and Germany, have said that they will continue to uphold it.
Trump made clear that there may be punishments for the UK and other European nations when he announced his withdrawal from the “decaying and rotten” agreement last Monday.
He said: “Any nation that helps Iran in its quest for nuclear weapons could also be strongly sanctioned by the United States.
“We will be instituting the highest level of economic sanctions.”
After the President threatened foreign firms that continue trading with Iran with “strong sanctions”, Trump’s national security adviser John Bolton told American TV he thought the UK and France would follow suit and ditch the 2015 deal.
He told CNN’s State of the Union: “The Europeans will see that it’s in their interests to come along with us."
When asked if this would mean sanctions against firms, the adviser didn’t deny the possibility.
He added: “It’s possible, it depends on the conduct of other governments.”
The Secretary of State has also refused to rule out sanctions against European firms.
Speaking to Fox News Sunday, Mike Pompeo said: “The sanctions regime that is in place now is very clear on what the requirements are.”
The Republican also accused the international agreement of encouraging “malign activity” in the Middle East.
The American government has already blocked US firms from doing deals in Iran, following the President’s rejection of the deal last week.
Foreign companies that do business in the country have also been banned from accessing American banking and financial systems.
The deal was signed in 2015 and was designed to limit Iran’s nuclear activities, in exchange for the lifting of harsh international economic restrictions.
Under the deal, signed by President Obama, firms had agreed billions of dollars worth of contracts with Hassan Rouhani’s government.
Iran has promised to maintain the deal, regardless of Trump’s withdrawal, as long as it still has the support of France and the UK.
Speaking last week, President Rouhani said: “If we achieve the deal’s goals in co-operation with other members of the deal, it will remain in place.”