The US has banned Iranian oil imports to its own shores for decades, so Washington's ability to squeeze the exports will rely on convincing foreign governments and firms to pressure Iran back to the negotiating table for a "better deal" - as Trump has said he expects.
"My expectation is not that oil prices go higher", he said.
Oil prices have been rising in the past few weeks due to worries over the possibility that the US could reimpose sanctions on Iran, which would then hinder Tehran's ability to export oil and create disruption in the global supply. That's about 1 million barrels a day more than in early 2016.
The investors are anxious that renewed sanctions against Iran, a major oil producer, could lead to supply disruptions.
Paul Sheldon, an associate director at Platts Analytics, said enforcing compliance with Trump's unilateral sanctions move is much more hard than previous multilateral sanctions, however.
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Hedge funds cut their net long U.S. crude futures and options positions in the week to May 1 by 11,825 contracts to 444,060, according to the U.S. Commodity Futures Trading Commission.
At least some of that oil will now be pulled from the market at a time when oil prices are already rising because of production cuts by the Organization of the Petroleum Exporting Countries (OPEC) and Russian Federation as well as instability in Venezuela, CNN Money reported.
The increased demand helps to overcome the over-crowding that has been pursuing the markets.
Under the announcement, companies selling commercial aeroplanes to Iran will be particularly hit.
Crude has been rallying ahead of an announcement by the Trump administration on whether or not the United States will remain in the Iran nuclear deal. Now Iran, the world's fifth-largest oil producer, will have to decide whether to follow the US and withdraw or try to salvage what's left with the European countries.
The Prime Minister raised the potential impact of U.S. sanctions on those firms which are now conducting business in Iran.