The Saudi Press Agency said Trump had called Salman and the two highlighted "the necessity of doing efforts in order to preserve the stability of the oil market and the global economy growth".
Saudi Arabia acknowledged the call took place, but mentioned no production targets, despite Trump tweeting in reference to Salman, 'He agreed!'
According to the report, the two leaders also spoke about efforts of oil producing countries to compensate for any potential shortage of supply.
"All the potential shortfalls could outstrip the production increase agreed to by OPEC and Russian Federation", said Dominick Chirichella, Director of Risk Management at EMI DTN.
The development came days after the oil-rich Persian Gulf state, which is the world's largest oil exporter, agreed to substantially raise production.
The U.S. government wants to stop Tehran exporting oil to cut off a vital supply of finance, and hopes other big oil producers in the Organization of the Petroleum Exporting Countries and Russian Federation will make up for the deficit.
An agreement by an OPEC member nation to increase oil output at the request of a USA president would be a serious blow to that alliance of 15 oil-producing countries, which sets production quotas collectively, and has already shown fragility in recent days.
The overt pressure brought by Trump on Saudi Arabia is unusual, but the kingdom has been actively urging the president to withdraw from the worldwide agreement that placed restrictions on Iran's nuclear program. Gen. Mohammed Barkindo (R) and Angola's Governor for OPEC and Chairman of the Board of Governors Estevao Pedro (2nd L) at the 173rd OPEC Conference in Vienna on November 30, 2017.
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However, Saudi Aramco CEO Amin Nasser told journalists in India on Monday that the state oil company has spare capacity of 2 million barrels of oil per day.
Meanwhile, UK-based energy consultancy group, Fact Global Energy (FGE), reported in its June newsletter that Washington is determined to "push Iran's exports to zero". Added to the losses from Venezuela, is there enough capacity to replace the lost Iranian volume??
Trump made the boast on Twitter on Saturday following a telephone call with King Salman in which the two discussed rising oil prices.
Benchmark Brent crude LCOc1 was trading around $79 a barrel on Friday, and a Reuters poll showed prices look to remain strong for the rest of this year due to supply disruptions in countries including Libya and Venezuela and as the extra oil from OPEC fails to meet rising demand.
The talks had centered on whether to amend an 18-month-old supply-cut deal between OPEC members and allied countries, including Russian Federation, that has cleared a global oil glut and lifted crude prices.
Iranian exports have been running around 2.4 million barrels a day, according to the International Energy Agency.
At the OPEC meeting last week, Iran's oil minister said that the cartel should not be influenced by Trump and should deal with supply and demand. Turkey stood at 9 percent and Italy at 7 percent.
By the end of this year, the call on OPEC is expected to be almost 1.5 million BPD more than they were forecast to produce.
Iran's Supreme Leader Ayatollah Ali Khamenei on Saturday accused Washington of trying to turn Iranians against their government.