President Bush said Tuesday Israeli Prime Minister Ehud Olmert's ideas to remove from the West Bank most Israeli settlements except for the primary population centers is "an important step toward the peace we both support."

Olmert said he is willing to relinquish much of the West Bank to the Palestinians. A border would be drawn by 2010. However, Olmert is looking for clues on whether the U.S. president will insist that any new map of the coming two states require the approval of Palestinian leaders.

Bush said the best solution is one on which both Palestinian and Israeli negotiators agree.

"I believe, and Prime Minister Olmert agrees, that a negotiated final status agreement best serves both the Israelis and the Palestinians and the cause of peace," the president said in a press conference with Olmert in the White House East Room.

But, Bush added, "No country can be expected to make peace with those who deny its right to exist and who use terror to attack its population."

Olmert said Bush's leadership in creating a two-state solution has been "significant."

"The vision which you outlined in your historic speech of June 2002 of two democratic states living side by side in peace and security is the basis of any progress toward a solution in this region," Olmert said.

During the press conference, Bush recalled the "great contributions to peace" made by Ariel Sharon, who preceded Olmert as prime minister before suffering a massive stroke in January. Sharon, a war hero dating back to 1967, first laid out the plan for unilaterally redrawing Israel's borders. The proposal created such a divide within the Likud, or "Consolidation," party that Sharon headed, that he left and created the Kadima, or "Forward," party, of which Olmert became his chief ally.

Prior to their press conference, Bush met in the Oval Office with Olmert. Bush said afterward the two leaders were going to retire to the president's residence and speak more, without the inclusion of their delegations.

Several sources told FOX News that the private meeting between the two leaders also focused on how to contain Palestinian-on-Palestinian violence in the Gaza Strip, and how to maintain communications with the Palestinian Authority, whose current leadership is designated by the U.S. State Department and European Union as a terrorist organization.

Hamas, which is now running the Palestinian government, refuses to accept Israel's existence or to renounce attacks on the Jewish state. But Israeli officials say they are willing to work with Palestinian Prime Minister Mahmoud Abbas, who has agreed to follow the road map to a two-state solution.

Bush said Abbas serves as the best conduit to creating a final peace agreement between Israelis and Palestinians.

"Prime Minister Abbas speaks out for peace and negotiations, yet the Hamas-led Palestinian government does not. Hamas needs to make a strategic choice for peace" by recognizing Israel's right to exist, abandoning terror and upholding prior peace agreements, Bush said.

"I intend to exhaust every possibility to promote peace with the Palestinians according to the road map, and I plan to extend my hand to Mahmoud Abbas, the elected Palestinian president," Olmert said.

"Despite our sincere desire for negotiations, we cannot wait indefinitely for the Palestinians to change. We cannot be held hostage by a terrorist entity which refuses to change or to promote dialogue. If we come to the conclusion that no progress is possible, we will be compelled to try a different route," he added.

Last weekend at the World Economic Forum in Egypt, Abbas appealed for an end to violence among Palestinians in Gaza. He also met with Israeli Foreign Minister Tzipi Livni.

But recent weeks have seen sharply escalating violence among Palestinians, bordering on a low-intensity civil war between security forces deployed by the Hamas-led government and armed factions loyal to Abbas and his Fatah party.

U.S. officials say they are harboring no illusions about Abbas' efficacy or Hamas' intentions. White House spokesman Tony Snow said before the Bush-Olmert meeting that dealing with Abbas is the logical solution since Hamas does not want to negotiate.

"We know that President Abbas has gone ahead and at least supported the conditions that we've always said are necessary for a two-state solution. On the Palestinian side, you're going to need somebody who recognizes Israel's right to exist, renounces violence, who recognizes prior agreements. Now, that is not the case right now with Hamas. It is the case with President Abbas," Snow said.

Snow also said he did not expect "anything formal" to emerge from the White House meeting.

"The meeting with Prime Minister Olmert really is, in many ways, a getting-to-know-you session. The president and the prime minister have known each other when they held different jobs, one a mayor, one a governor. And while they have struck up something in the way of personal relations, this is a chance to get to know each other better but also roll up their sleeves and talk about the best ways to move forward, once again, a two-state solution so that we can try to create the conditions for peace in the region," Snow said.

Olmert said that Israel's plan would be to "remove most of the settlements which are not part of the major Israeli population centers in Judea and Samaria." That action would reduce friction and help guarantee Israel's security, he said, but such a plan would need the support of the United States and the international community.

Olmert began his Tuesday in Washington with an early morning courtesy call on Defense Secretary Donald Rumsfeld at the Pentagon. A spokesman there said the two discussed Iran but did not cover its nuclear program in detail.

The 35-minute meeting also touched on Iraq and U.S.-Israeli technology sharing.

Iran was taken up as "a country of concern that we have to pay attention to," Pentagon press secretary Eric Ruff said. Olmert and Rumsfeld agreed to continue to exchange information about Iran, Ruff said.

Speaking to reporters at the White House, Olmert noted that the Iranian regime denies the Holocaust, views the United States as its enemy and makes every effort to implement its fundamentalist religious ideologies in blatant disregard of the demands of the international community.

"The Iranian threat is not only a threat to Israel; it is a threat to the stability of the Middle East and the entire world. And it could mark the beginning of a dangerous and irresponsible arms race in the Middle East," he said.

Bush said that an attack by Iran on Israel will not be tolerated, though he didn't specify what measures the United States would take to defend the Jewish nation.

"I told the prime minister what I've stated publicly before: Israel is a close friend and ally of the United States. And in the event of any attack on Israel, the United States will come to Israel's aid. The United States is strongly committed, and I am strongly committed, to the security of Israel as a vibrant Jewish state," Bush said.

Sources told FOX News that the Bush administration is prepared to accept European calls for some limited foreign payments to Palestinian health and education workers, but not a massive cash flow to the Hamas-led government.

The White House is also ready to go along with the EU's idea for vouchers that would amount to food stamps. However the administration is opposed to a massive cash flow to the PA that could wind up in the hands of the Hamas security forces.

Though the White House has already cut off funds to the PA, the administration's position conflicts with the House, which voted overwhelmingly on Tuesday to ban U.S. assistance to the Palestinian Authority and any diplomatic contacts with Hamas while also denying visas to members of the PA.

The chief sponsor, Rep. Ileana Ros-Lehtinen, R-Fla., said it would send a message to Hamas that the United States will not support a government run by terrorists.

The Senate is considering a less restrictive bill. Snow said Bush does not support the House measure "precisely because it does tie the president's hands" in regard to providing humanitarian aid and "unnecessarily constrains" the administration. He added that president is hoping a House-Senate conference will make the bill more palatable.

Last night, Olmert and his top aides had dinner with Secretary of State Condoleezza Rice, Assistant Secretary of State for Near Eastern Affairs David Welch, National Security Adviser Steven Hadley and the National Security Council's Elliott Abrams.

Sources said the dinner amounted to a "rehearsal" for the Bush-Olmert meeting. A source described the session as "substantive ... with very warm tones."

Olmert is meeting Wednesday with Vice President Dick Cheney and House Speaker Dennis Hastert, R-Ill. and will speak to a joint meeting of both chambers of Congress.

FOX News' James Rosen and The Associated Press contributed to this report.