Outside North America, looming US sanctions against major oil exporter Iran were the focus of attention.
"We are prepared to work with countries that are reducing their imports on a case-by-case basis, but as with our other sanctions, we are not looking to grant wavers or licenses", Hook said when asked about India and Turkey which import Iranian oil.
Last week, US State Department officials announced that they are pushing nations to bring down purchases of Iranian oil to zero.
The official Saudi Press Agency confirmed a phone call between Trump and Salman about oil, but mentioned no specifics and the White House later struck a more cautious tone, saying in a statement the two leaders had "reaffirmed their dedication to a healthy and stable global energy market".
"And so, we are not looking to grant licenses or waivers broadly on the reimposition of sanctions, because we believe pressure is critical to achieve our national security objectives", he said.
The spike was the latest fallout from the Trump administration's decision to abandon the landmark Iran nuclear deal and reimpose sanctions on Iran - a move that has strained relations between the United States and other world powers that support the deal.
The planned production increase was meant to balance the market as global oil prices have been steadily rising, edging closer to $80 per barrel last month.
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The president, in an interview with Fox News's "Sunday Morning Futures with Maria Bartiromo also piled pressure on close United States allies with a threat to sanction European companies that do business with Iran".
Mr Trump took another swing at his European allies saying they dealt unfairly with the U.S. over trade.
Outside North America, looming USA sanctions against major oil exporter Iran were the focus of attention.
He said in the Fox interview that European companies would face sanctions if they traded with Iran now.
"The Iranian government has a plan. and God willing we are certain that we will be able to sell as much oil as we want", he said. "Iran. OK. You think of it. Iran is their big enemy, so they are going to have to do it", Trump said.
Oil prices rose strongly last week, with the U.S. crude contract hitting its highest in 3-1/2 years at $74.46.
Saudi Arabia's output is up by 700,000 barrels per day (bpd) from May, a Reuters survey showed, and close to its 10.72 million bpd record from November 2016.