OPEC countries and their 11 allies, including Russian Federation, agreed at a scheduled meeting in June to return the production to a 100% compliance with the terms of the output reduction deal.
The White House later backtracked on the president's tweet, while Saudi Arabia did not confirm Trump's comments.
The cabinet commended "constructive cooperation between producing countries from inside and outside OPEC, which resulted in the agreement of 25 countries to increase oil supplies, taking into account the current market conditions".
Hook declined to say if the kingdom can offer that additional supply, but the United States is "working to minimize disruptions to the global market", Hook said.
Also on Thursday, data showed USA crude stockpiles rose unexpectedly last week by 1.2 million barrels.
United States officials are pressing allies in Europe, Asia and the Middle East, including Saudi Arabia, to adhere to the sanctions, which are aimed at pressuring Iran to negotiate a new agreement to halt its peaceful nuclear programmes.
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Oil prices rose on Friday on worries that U.S. sanctions against Iran would take away significant volumes of crude oil from world markets while oil demand worldwide increases.
Despite the apparent supply relief from Saudi Arabia, oil markets remain tense over escalating trade disputes between the United States and other major economies including China, the European Union, India and Canada, as well as the looming new USA sanctions against Iran.
The drop is in contrast to the strong growth in oil prices seen last week as prices started to nudge close to $80 per barrel. But those moves may be tempered by an equal desire by the USA and other country for stable oil prices. Average pump prices in the US are up 13 percent this year, rising above $3 a gallon in April for the first time since 2014.
China is the largest importer of Iranian oil with 24 percent, followed by India with 18 percent.
Oil held near US$74 a barrel amid estimates that U.S crude inventories declined for a fourth week, compounding concerns that global markets are growing increasingly tight.
"Oil is already being offered on the bourse, about 60,000 barrels per day, but that has been only for exports of oil products", Jahangiri said.
U.S. officials from the Treasury Department and State Department have been taking what Hook referred to as travelling "road shows" to explain the policy to their counterparts around the world over the past few weeks, ahead of August 4, when the first sanctions will be reimposed on automotive, gold, and key metals.
The world's biggest crude exporter is preparing to ramp up oil production, possibly by as much as two million barrels per day, amid USA sanctions on Iran.