The Saudi Press Agency said Trump had called Salman and the two highlighted "the necessity of doing efforts to preserve the stability of the oil market and the global economy growth".
The Democratic opposition has blamed part of that increase on Trump's decision to withdraw the USA from the 2015 nuclear deal with Iran, while Republicans fear that rising gasoline prices will tiresome Americans' enthusiasm for the U.S. economy as they approach next November's legislative elections.
"President Trump's reckless decision to pull out of the Iran deal has led to higher oil prices", Senate Minority Leader Charles Schumer, D-N.Y., said last month during a photo op outside an Exxon station. US prices for retail gasoline at the moment have reached levels that haven't been matched in the past four years.
Oil analysts said OPEC producers may not be able to fully supply the market if Iranian oil is cut from the market.
In case of need Saudi Arabia can use the available reserves to increase production of oil, is in conversation with the President of the United States Donald trump promised Saudi Arabia's king Salman, the White house said.
Saudi Arabia's pledge followed a meeting earlier this month in Vienna of OPEC members, Russian Federation and other oil producing countries.
The telephone exchange is another sign of how US-Saudi ties have improved under Trump compared with the Obama administration, which alienated the kingdom by seeking a nuclear deal with Iran.
If the Saudis had agreed to Trump's request, "that means he is calling on them to walk out from OPEC", Iran's OPEC governor Hossein Kazempour Ardebili, said in an interview.
The US government has threatened with sanctions all the companies in the world that starting next November 5 continue doing business with Iran, which includes oil purchases. "Prices to high! He has agreed!", Trump said on Twitter Saturday.
There was no immediate comment from Saudi authorities.
It added that there also was an understanding that oil-producing countries would need "to compensate for any potential shortage of supplies".
"This will push the system close to the limit", and spare production capacity could fall below 2 million barrels a day - a level previously associated with $100 per barrel prices, the group said.
Saudi Energy Minister Khalid al-Falih met with US Secretary of State Mike Pompeo in Washington on Thursday to discuss energy security. The kingdom pumped slightly more than 10 million barrels a day in May.
Iran had been pushing hard for oil producers to hold output steady as USA sanctions are expected to hit its exports. "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. Estimates of how much spare capacity or quickly accessible oil is available vary, but most experts think it would be challenging to cover a large drop in Iranian exports combined with other outages.