Washington — Chevron's chief executive predicted travelers could see airline fares rise and fewer flights available in the coming weeks, as the standoff with Iran over the Strait of Hormuz continues to rattle global markets and drive up the price of fuel. 

"We've seen some upward pressure on gasoline prices now. I think aviation is clearly an area where it's going to probably get worse over the next few weeks," Chevron CEO Mike Wirth said in an interview Thursday with "Face the Nation with Margaret Brennan." 

"We are seeing jet fuel tighten very quickly in Europe, in Asia, and we're seeing airlines announce adjustments in their flight schedules. We're seeing it flow through into fares. I think that's one of the first places it will be felt most broadly," he said. 

Wirth said there was already a jet fuel shortage in certain parts of the world before the war with Iran began on Feb. 28. Airlines have hiked their bag check fees and cut routes since the start of the war in response to the crisis. Carriers based in the U.S. are slightly better positioned than European airlines because the U.S. produces its own jet fuel. 

"I think the upward pressure that they're seeing on prices and the tightness in the market is likely to lead to further route optimization. And so flights may not be as abundant as they otherwise would have been. I think planes will probably be more full than they would have been," Wirth said.

"And yes, fares — fares could be higher," he added. 

In North America, jet fuel prices have surged more than 80% compared with this time last year, according to the International Air Transport Association. 

The average cost of gas nationwide was $4.03 per gallon on Thursday, approaching nearly a dollar more than the price per gallon a year ago, according to CBS News' price tracker. Diesel, which powers trucks, boats and trains, has risen more quickly than regular gasoline and was at $5.47 per gallon on Thursday. 

Watch more of the interview Sunday on "Face the Nation with Margaret Brennan."