Trump could withdraw the US from the historic nuclear deal with Iran, and impose sanctions on the country, which then would make it harder for Tehran to export crude and oil products to the global market. Meanwhile, industry data was said to signal an unexpected decline in American crude inventories last week.
For the record, during that meeting on Tuesday, Salman told the brash billionaire he sees a "stable" oil market ahead and added that his kingdom has an 84-year supply of crude.
The specter of conflict involving giant producers is jolting prices, which have traded in a $4-range since mid-February.
U.S. West Texas Intermediate (WTI) crude futures were up 87 cents at $62.93 a barrel.
Bearish concerns have largely been fueld by surging US crude output.
However, all along, there was the implicit concern of the activities of shale producers in America whose new technology coupled with favourable government protectionist policies in the oil sector have helped the United States raise production from 8 million over the last decade, to 10.38 million barrels per day (bpd), pushing it past top exporter Saudi Arabia.
Brent crude, the global benchmark, was up almost 1% at $66.69 a barrel on London's Intercontinental Exchange.
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The US trade deficit with China ran to a record US$375 billion past year - but US exports to the country were also at a record. He said there would be no victor in a trade war, and that both sides should remain rational and avoid giving in to emotions.
Refinery crude runs rose 410,000 bpd and refinery utilization rates jumped 1.7 percentage points to 91.7 percent of total capacity, EIA data showed. Brent is the benchmark for several Middle East and other global crudes.
While US energy demand is at 11-year highs, US crude oil production is also surging at the same time.
If Trump decides to exit the deal under which global measures were removed on Iran in return for a curbing of its nuclear program, it would likely be welcomed by Saudi Arabia, which regards the deal as a boon for its regional foe.
Energy consultancy FGE said it was likely that the United States would re-impose sanctions on Iran soon, resulting in a 250,000 to 500,000 barrels per day (bpd) drop in its exports by year-end.
"I think there are conflicting views on the USA crude stockpiles and we should wait for the EIA data to confirm the decline", said Hyundai Futures' Yun.
Many analysts expect global oil markets to flip from slight undersupply in 2017 and early this year into oversupply later in 2018.