Oil production by the countries of the Organization of the Petroleum Exporting Countries has dipped because of supply disruptions in Iran, Venezuela and Angola, driving prices to near $80 a barrel.
OPEC agreed with Russian Federation and other oil-producing allies this month to raise output from July by about 1 million bpd.
"There is no way one country could go 2 million barrels a day above their production allocation unless they are walking out of OPEC", he said.
Brent price on Friday rose $1.54 a barrel to $79.39 on concerns that U.S. sanctions against Iran would remove a substantial volume of crude from world markets at a time of rising global demand.
The state-run Saudi Press Agency confirmed Saturday that Trump had spoken with Salman on the phone about the need to "maintain the stability of oil markets", The Wall Street Journal reported.
On Saturday, Trump said that he spoke to the Saudi king about the need to increase Saudi Arabia's oil production "maybe up to 2,000,000 barrels" due to unstable situation in Iran and Venezuela, stressing that the Saudi leader agreed with the proposal.
The curbs were meant to help drain a global oil glut, a goal that has largely been achieved, though supply disruptions are now adding pressure to prices.
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Still, precise details about the agreement were in shorter supply than leaders' expressions of success on Friday. Central European countries claimed they had buried the idea of mandatory refugee quotas for all EU countries.
Mr Trump has repeatedly criticised Opec even though USA ally Saudi Arabia is a core member.
"Prices to [sic] high!" Average prices for unleaded gas, including taxes, averaged $2.83 a gallon for the week ending June 25 - up about 55 cents over the same period past year.
FGE, an energy consultancy, said in its June newsletter that the U.S. is determined to "push Iran's exports to zero". If we assume a modest further decline in Venezuela's production and add Iran's likely loss of oil exports, then about one million bpd of additional oil would be needed to fill the gap.
It was unclear whether the Saudi king "agreed" to increase production or "agreed" prices were too high. Its record is 10.72 million barrels a day. Trump's claim of a 2-million-barrel increase for Saudi Arabia alone does not specify a timeframe.
The administration has threatened close allies such as South Korea with sanctions if they don't cut off Iranian imports by early November.
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. In Libya, where a dispute over control of key ports has hindered output, the Arabian Gulf Oil Co. on Saturday halted 220,000 barrels a day of production, according to a person familiar with the outage.
The oil markets were also shaken last week when a senior State Department official said that the United States would ask countries to reduce their purchases of Iranian oil to "zero".