Investing.com -- Chevron Corp. Chief Executive Officer Mike Wirth expressed concern about the security of ship transit through the Strait of Hormuz as the US works to restore passage through the Persian Gulf waterway.

The US and Iran exchanged fire in the region on Monday as American military forces began escorting vessels through the strait. The escalation followed US President Donald Trump's plan to reopen Hormuz and threatened a fragile ceasefire declared last month.

"I've been in touch with our people that manage our shipping assets in the region and we remain concerned about the security of transit," Wirth said during a Bloomberg TV interview at the Milken Institute Global Conference. "So it seems like we still have some issues to work through."

Chevron and other major oil companies have reduced Middle East production due to conflict that began at the end of February. The war closed the strait, affecting about 1 billion barrels of global oil cargoes and pushing international crude prices up almost 60% in roughly nine weeks.

Refiners have been using barrels held in storage, which may face eventual depletion. Wirth said he advised the Trump administration that the supply situation is tightening.

"I've advised people in the administration that the buffers in the system that help ensure supplies are available to markets are being drawn down," Wirth said. "It creates more upside price pressure, potentially more volatility and more risk."

As the world's largest oil producer, the US is unlikely to see fuel shortages but consumers will see price increases, Wirth said. The risk of supply outages in other parts of the world is higher than in the US, he added.