Jan. 16, 2008  -- One hour after his plea for more Saudi oil was publicly rejected by the kingdom's oil minister, President Bush made a private visit to Saudi Arabia's King Abdullah to again ask him to open the spigots.

The White House revealed Bush's private meeting with the Saudi monarch to reporters aboard Air Force One as the president flew to Egypt on the next leg of his Mideast trip.

Earlier Tuesday, Bush made his case for having OPEC, and particularly American ally Saudi Arabia, increase oil production as the price of gas hovers around $3 a gallon.

The Saudi oil minister, however, waited only a short time before announcing that oil prices would remain tied to market forces — a direct slap at Bush.

The president went over the head of the oil minister and made his case to King Abdullah and White House Press Secretary Dana Perino said the private conversation may have yielded some daylight in the Saudis' hard-line stance.

"The king says that he understands the situation. He's worried about high oil prices and how they can negatively affect economies around the world," Perino said aboard the presidential jetliner. "The president said there's a hope that as a result of these conversations that OPEC would be encouraged to authorize an increase in production … to help deal with the tight supply problems in this time when we have growing economies across the world, especially in China."

Earlier, Bush told ABC News' Terry Moran how he would lobby the king.

"I will say to him that, 'If it's possible, your majesty, consider what high prices are doing to one of your largest customers,'" Bush said. "In other words, the worst thing that can happen to an oil-producing nation is that the price of oil causes the economy to slow down, because that will inevitably lead to fewer purchases [of oil]."

Bush said he's worried about an economic slowdown in the United States and around the world because of those high oil prices.

"These are smart people. They know that the price of oil can affect our economy, and they know that if our economy weakens and there's less purchasing power, that it will affect their ability to sell barrels of oil," the president said.

Perino added, "This is not a situation that's going to be solved overnight, and that's why on a parallel track the president has been pursuing aggressively alternative and renewable forms of energy."

When Bush ran for the White House in 2000, he said he would "jawbone" America's Saudi allies to lower the price of oil. The price of oil has hit a record of $100 a barrel, but has slipped to a current price of $91 a barrel on fears that the U.S. economy is headed for a recession.

New York Sen. Hillary Clinton took a shot at Bush's requests to the Saudis during the Democratic debate in Nevada Tuesday night. Referring to his pleading, she called the tactic "pathetic."