The US president made the announcement on Sunday, as he left for his retreat at Camp David to discuss options concerning the Middle Eastern country.
Tensions in the region are high after the US accused Tehran of attacking oil tankers in the Gulf of Oman and shooting down an unmanned drone.
“We are moving forward with additional sanctions on Iran,” Mr Trump told reporters. “They’re going on slowly and in some cases pretty rapidly.”
The president stressed military action was still a possibility, after he called off a planned strike, but said he wanted the response to be “proportionate”.
Mr Trump said it was important to “start over on Iran” and added a riff on his regularly-repeated slogan: “Let’s make Iran great again.”
He went on to say that hopefully Tehran would be “smart” about the conflict and “cares about its people”, adding that he had a number of Iranian friends who were “high quality people”.
Mr Trump also noted that while his security adviser, John Bolton, was “definitely a hawk”, he intended to listen to every perspective on dealing with the problem.
In an interview with NBC’s Meet the Press, the president said a war with Iran would cause “obliteration like you’ve never seen before”, but said he had no desire for armed conflict.
“I’m not looking for war, and if there is, it’ll be obliteration like you’ve never seen before. But I’m not looking to do that,” Mr Trump said.
The comment was reminiscent of his "fire and fury" rhetoric directed at North Korea, which softened once he and Kim Jong-un got to the negotiating table.
It came as Iran reportedly executed a former employee of its defence ministry on charges of spying for the CIA.
Documents and spying equipment were found at Jalal Hajizavar’s home, IRIB News reported.
Mr Hajizavar, who had left his position as an aerospace contract employee for the ministry nine years prior, was convicted by a military court and sentenced to death.
Tensions between the US and Iran have escalated after Mr Trump pulled out of the Iran nuclear deal.
Mr Trump claimed the US was “cocked and loaded to retaliate” after Tehran shot down a US drone but decided against a strike when he was informed as many as 150 people could be killed.
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