U.S. oil production likely hit a record of 12.5 million barrels per day in May and should even grow to 13.4 million barrels daily by the end of this year, the Norwegian research firm Rystad Energy projects. Next year should see oil supplies hit 14.3 million barrels a day.

"U.S. oil production is already higher than many in the market believe," says Bjørnar Tonhaugen, Rystad's head of oil market research.

"Strong growth persists in the Permian Basin on both the New Mexico and the Texas sides," he said. "Updated production estimates suggest that the Permian Basin surpassed 4.5 million barrels per day in May."

Considering the ongoing recovery in hydraulic fracturing, called fracking, activity, the ship isn't turning around anytime soon even as oil prices have turned bearish at the end of May and into June, Tonhaugen added.

The U.S. Energy Department has projected that U.S. oil output is currently at a still-record 12.4 million barrels a day, but that the May average was less than 12.3 million barrels daily.

The government estimates the Permian's current volumes at less than 4.2 million barrels of crude oil a day, well less than Rystad's analysis.

Rested believes the Permian will exceed 5 million barrels daily at some point next year.