WASHINGTON – When Prime Minister Sharon visits America next week to meet with President Bush, their joint understanding of the Israeli-American relationship may be tested. At stake is a final settlement to the Jewish state’s conflict with the Palestinian Arabs.

The Israeli leader yesterday told members of the Knesset that a proposed road connecting Jerusalem to the town of Maaleh Adumim must be built. “We must link Jerusalem to Maaleh Adumim,” Mr. Sharon said to lawmakers, Reuters reported.

Construction of the road, along with proposed expansion of housing units in the Jerusalem suburb, will be a powerful test of Mr. Bush’s belief that a final agreement on the Arab-Israeli conflict ought to recognize the demographic realities of Jewish settlements.

On April 14, 2004, Mr. Bush wrote to Mr. Sharon that it is “unrealistic to expect that the outcome of final status negotiations will be a full and complete return to the armistice lines of 1949.” Both houses of Congress quickly passed resolutions endorsing the letter.

Israeli leaders have interpreted the understandings in the April 14 letter to mean that America accepts – and indeed supports – a final settlement with the Palestinian Authority that would leave Jewish settlements like Maaleh Adumim within Israel’s borders.

But in recent weeks, some American senior officials have criticized Israel’s recent efforts to expand Maaleh Adumim. The Jewish settlements around Jerusalem are expected to come up Monday in Crawford, Texas, at Mr. Bush’s ranch. Next week’s parley will also provide the two leaders an opportunity to discuss the Israeli withdrawal from Gaza scheduled for July 20.

In the last week Israelis have voiced concerns that Palestinian leader Mahmoud Abbas has failed to take seriously his commitments to dismantle the three major terrorist groups operating in Gaza and the West Bank. Some Israelis have complained specifically that wanted criminals remain at large in towns from which Israeli forces are scheduled to withdraw in coming weeks.

“In the final analysis the negotiation of the permanent agreement will reflect the new realities, the new demographic changes will have to be taken into account. Israel will not be expected to pull out of those areas,” Israel’s Vice Prime Minister, Ehud Olmert, told reporters yesterday in a conference call. He added that he believed Mr. Bush’s April 14 letter on this matter was “crystal clear.”

In the past few weeks, however, the administration’s policy on the issue has become murkier. A State Department official yesterday told The New York Sun that Secretary of State Condoleezza Rice and National Security Adviser Stephen Hadley pressed a senior adviser to Mr. Sharon, Dov Weisglass, for “further clarification of the prime minister’s statements” on Maaleh Adumim.

Ms. Rice last month told the Los Angeles Times, “We expect, in particular, that they are going to be careful about anything, route of the fence, settlement activity, laws, that would appear to prejudge a final status agreement, and it’s concerning that this is where it is and around Jerusalem.”

The executive vice president of the Conference of Presidents of Major American Jewish Organizations, Malcolm Hoenlein, yesterday said, “The administration has reiterated its commitment to Israel retaining the major population centers, which have to include Maaleh Adumim, a city of over 20,000 people.”

A senior fellow at the Council on Foreign Relations and former executive director of the American Jewish Congress, Henry Siegman, disagreed with Mr. Hoenlein’s interpretation. “I think it is difficult to construe the April 14 letter as a carte blanche for Israel to proceed with the expansion of the existing settlements,” he said. “In fact President Bush has said specifically that the expansion of settlements violates existing agreements.” Mr. Siegman said yesterday that the proposed road further cuts off parts of Arab East Jerusalem from the rest of the territory that the international community envisions will be part of a Palestinian state.

The status of Jerusalem bedeviled American-sponsored negotiations between a former Israeli prime minister, Ehud Barak, and the late Yasser Arafat. Mr. Olmert yesterday said, “Every inch of land between the Jordan River and the sea is part of the Jewish heritage. It has always been.” He added that the current dilemma for Israel rests ultimately in how to manage the demographic facts of a Palestinian Arab majority in the West Bank territory. “Do we want a binational state with complete equality, or do we want to have a state that is predominantly inspired by the Jewish heritage?” Mr. Olmert asked. “We can’t have both. I think the Palestinians are entitled to have an independent state, and we are entitled to have our independent state. … There will have to be some sharing.”