Notable countries that import Iranian crude include Turkey, India, China and South Korea.
After the remarks, US crude jumped more than 8 percent and closed at more than $74 a barrel - a first since November 2014. India has said they intend to comply, and even if China doesn't, it's not clear they'll be buying enough to prevent these sanctions from being a major challenge to the markets.
Iran will use various methods to prevent a possible reduction in its oil exports after re-imposition of the US sanctions.
Rouhani spoke to Iranian expatriates in Switzerland during his visit there on Monday.
Iran vowed Sunday to defeat USA efforts to block its oil exports and warned rival producer Saudi Arabia it would never take Tehran's "place" on the worldwide oil market.
"Our alternative ready-made preference will be Iraq, Saudi Arabia and the UAE due to proximity". But the Iranian government has said it will not continue to abide by the agreement if doing so goes against its economic interests.
Oil prices fell on Monday, reversing course from last week as supplies from Saudi Arabia and Russian Federation rose while economic growth stumbled in Asia amid escalating trade disputes with the United States.
Oil prices have been rising despite OPEC and Russian Federation agreeing to increase production by 1 million barrels per day from the start of this month.
US officials are pressing allies in Europe, Asia and the Middle East to adhere to the sanctions once they are re-imposed, with the aim of pressuring Iran into negotiating a new agreement.
"We're not granting waivers", the official said. He said it is not possible that Iran will not be able to export its oil while others in the region will be free to do that.
"In this battle, any country that tries to take Iran's place on the oil market will be guilty of treason against Iran. and surely one day it will pay the price of this treason", he said. "Prices to high! He has agreed!" But those moves may be tempered by an equal desire by the U.S. and other country for stable oil prices.
The White House has since walked back the claim that Salman agreed to Trump's request.
"We are working to minimize disruptions to the global market but we are confident there is sufficient global spare oil production capacity", Brian Hook told reporters.