Pumped further by global trade war fears, oil prices have since surged to multi-year highs, only to then fall back again somewhat, the IEA said. "Not good!" the president said. American oil production has jumped by nearly a third in the last two years to a record 10.8 million barrels per day.
"Statements by several parties suggest that action in terms of higher supply could be on the way", said the IEA, which represents the United States and other oil-consuming nations.
OPEC and other producers will meet on June 22-23 in Vienna to discuss future production.
The IEA said it expects global oil demand to grow 1.4 million bpd this year, and in 2019, and will top 100 million bpd in the fourth quarter of 2018.
A mix of new US sanctions on Iran, chaos in Venezuela and robust global growth has tightened the oil market.
"More oil from OPEC plus is the base case", said Bjarne Schieldrop, analyst at Swedish bank SEB.
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"The president's tweets are really telling the other oil-producing countries to step up your production to make up for the coming sharp declines that are expected in Iranian oil exports", Yergin said. Prices have risen around 60 percent over the previous year.
"Over time it would have happened anyway because of the cutbacks in (drilling) investment, but definitely OPEC's cut in production helped speed the reduction of the oil glut", said Phil Flynn, an oil analyst for The Price Futures Group.
Oil prices reversed losses, USA crude settling 0.4% higher at $66.64 a barrel, and Brent crude rising 1.1% to $76.74, amid reports that Russian Federation wants a full-rollback of production cuts it had agreed to make with OPEC in 2016. Saudi Arabia has already started pumping more crude.
"Oil prices have come up a lot, but we're not near $4 gasoline", Bordoff said. "The real catalyst for the recent rise in prices is the sanctions on Iran".
In the letter, the senators said a run-up in oil prices is "effectively a tax on every American family's discretionary budget" since the price of oil directly affects the price of gasoline.
The country boosted production after the US lifted sanctions related to Iran's nuclear program in 2016, but analysts expect output to fall when the Trump administration's decision to withdraw from the deal takes full effect later this year.
OPEC itself spotlighted USA output in its own monthly report on Tuesday, citing the growth of non-OPEC supply as one of several question marks hanging over the situation.