CRAWFORD, Texas, April 24 - When he meets at his ranch here on Monday with Crown Prince Abdullah of Saudi Arabia, President Bush will confront one of his trickiest diplomatic relationships. He will look for help on oil prices, try to find common ground on the Arab-Israeli conflict and prod the crown prince to allow more democracy at home , even as both sides struggle with deep strains set off by the involvement of Saudis in the Sept. 11, 2001 attacks, American officials and analysts said.

Three years after the two leaders held a tense meeting here, American officials and analysts say the atmosphere has improved. But Mr. Bush remains under pressure from members of both parties in Congress to take a hard line with the Saudis when it comes to terrorism and Israel, they said, while the Saudis remain resentful that they do not get more credit for what they see as intensive efforts to be helpful and responsive to the United States.

American officials give the Saudis some credit for stepping up their efforts to combat terrorism within their borders and to stop the flow of money from Saudis to terrorist groups. They said Saudi Arabia was also helpful in recent months in pressuring Syria to begin withdrawing from Lebanon.

At the same time, the Saudis have been heartened by Mr. Bush's stepped-up involvement in seeking peace between the Palestinians and Israelis, analysts said. Now Mr. Bush will be seeking help from the Saudis to help bring down crude oil and gasoline prices, which are taking a toll both on the economy and on Mr. Bush's approval ratings.

"Since the Sept. 11 attacks, you've had this growing chorus of anti-Saudi voices in the United States, and not from the fringes," said Flynt L. Leverett, a senior fellow at the Saban Center for Middle East Policy at the Brookings Institution and former senior director at the National Security Council under Mr. Bush. "From the Saudi point of view, this administration, while not joining the anti-Saudi perspective, has not done much to quiet it. But now that Bush has been re-elected, they're in a sense willing to give the administration another shot at improving the quality of the relationship."

Their meeting is unlikely to result in any big breakthroughs. When it comes to oil, the Saudis have less ability to drive down global prices by increasing output than at many times in the past, because they are already pumping closer to their maximum sustainable capacity than during past price spikes.

But in part because of the growing domestic political pressure on Mr. Bush to show that he is doing everything possible to help bring down crude oil and gasoline prices, oil issues will play a more prominent role at this meeting than at the previous one here, in 2002.

"This is a new ballgame," said Robert W. Jordan, a former United States ambassador to Saudi Arabia. He said he expected Mr. Bush "to make a pitch to Saudi Arabia to expand production."

When the two leaders last met at the Bush ranch, the average price of a gallon of regular unleaded gas was $1.41, according to the American Automobile Association. The price is now about 50 percent higher.

Over the same period, global demand for oil has risen sharply, fueled by consumers in China and India. In 2002, Saudi Aramco, the state-owned oil company, says it produced 6.8 million barrels of oil per day. The Saudis are now estimated to produce about 9.5 million barrels a day.

As demand and production have increased, the Saudis have been left with less spare production capacity to smooth out price spikes. Three years ago it was about three million barrels a day, according to Saudi Aramco. Today "the cushion is very, very small," about 1.2 million barrels, according to Nawaf Obaid, an oil and security consultant.

The drop in Saudi spare capacity is a major reason why the Saudis are less able to influence the price of oil today, according to Mr. Jordan and Mr. Obaid.

The Saudis have announced new development plans. Their aim is to offset declines in older fields and to increase overall capacity to 12.5 million barrels.

Refineries are operating at more than 90 percent of total capacity in the United States. Still, gasoline prices have not risen as fast as crude oil prices in the last three years. Crude oil traded at $26.36 the day of the 2002 summit meeting, about half the current price.

Meanwhile, the Saudis have shifted away from their historical dependence on American oil giants, who once controlled Aramco. Six months after the 2002 meeting here, the crown prince complained in a private meeting with American diplomats that the American companies were being too greedy in their negotiations over a plan to give foreign investors access to some of Saudi Arabia's natural gas resources, American and Saudi officials said.

In relations between the United States and Saudi Arabia, much rides on personal relationships because the diplomacy tends to be conducted at the highest levels. Vice President Dick Cheney met Sunday in Dallas with the crown prince and will be at the Bush ranch on Monday, as will other top officials, including Fran Townsend, the White House's homeland security adviser, who has often been an envoy to the Saudis on terrorism and security issues.

Analysts said the Saudis remained suspicious about Mr. Bush's intentions when it comes to a final agreement between the Israelis and Palestinians. They said the Saudis were concerned about a number of developments in the Middle East, including the growing Shiite influence in the region, especially in Iraq, where the continued instability is a worry to the Saudis. Saudi Arabia's religious establishment is dominated by the Wahhabi branch of Sunni Islam.

Members of both parties in Congress are pressing Mr. Bush to do more to compel the Saudi leadership to crack down on clerics who are inciting anti-American and anti-Western feeling within Saudi Arabia, and to take more concrete steps to ensure that money given to charity in Saudi Arabia is not making its way to terrorist groups.

"The alliance between the U.S. and Saudi Arabia in terms of fighting terrorism is still a work in progress," said Representative Sue Kelly, Republican of New York, who met recently with Abdullah in Riyadh.

Richard W. Stevenson reported from Crawford for this article, and Jeff Gerth from Washington.

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