The UN chief is expected to deliver a message to the king from Olmert, who believes that the Saudi peace initiative is "good and challenging" and could serve as a basis for a dialogue, excluding a number of reservations.
In their meeting Monday, Ban told Olmert about the Quartet's plan to convene a summit between Saudi Arabia, Israel and moderate Arab countries in order to advance the peace process based on the Saudi initiative.
"If I receive such an invitation, I will consider it positively," the prime minister replied with a smile.
The UN secretary-general will convey this message to King Abdullah, as well as Olmert's praises for the king's courage and decisiveness.
"The Saudi initiative points to the kind's leadership," the prime minister said.
Olmert does not accept the clause in the Saudi initiative discussing the right of return, but Quartet representatives made it clear on Monday that a dialogue could be launched in order to decide on the agreements and leave the disagreements for the end.
The UN chief is expected to tell the Saudi king and the Arab leaders that Olmert is furious over the fact that the kidnapped Israeli soldiers have yet to be released, and that he made it clear that if there was dramatic development in that area he would find it difficult to convince the Israeli public that there was room for peace initiative.
Ban told Olmert that he understood the concern over security issues, but demanded that Israel fulfill its part in the Road Map and ease the Palestinians' suffering.
The UN secretary-general decided with restrained determination to clear his schedule for the coming days and focus on an effort to bring about a diplomatic breakthrough and a dialogue for peace between Israel and Arab states. Time is running out, he believes, and it is time to take action. He plans to bring the United Nations into the picture as a fair mediator.
Signs of change
A day after visiting a refugee camp on the eastern part of the "separation wall" and being exposed to the Palestinians' distress, Ban arrived Monday at the Yad Vashem, where he said that the world be on guard so that the horrors of the Holocaust do not repeat themselves.
He flew over Israel's "narrow waists" by helicopter and understood the importance of the "security fence" in thwarting terror attacks.
"I learned a lot," he said at the end of the day. After being exposed to the suffering on both sides, he decided to try and write a new episode in the region's history. Talking to Israeli and Palestinian leaders, he made it clear that he won't take "no" for an answer.
"There is new energy in the Middle East, and we must not lose the momentum. My partners in the Quartet and I now plan to pressure all the parties in order to advance the Saudi initiative," he said.
During a lunch with Acting President Dalia Itzik, the UN secretary-general met with the families of the kidnapped soldiers and promised to not let go until results are reached.
"I have details, but I cannot discuss them yet. It's very sensitive," Ban said.
Foreign Minister Tzipi Livni told him Monday evening that Lebanon was not fully implementing Security Council Resolution 1701, calling for the release of kidnapped soldier Ehud Goldwasser and Eldad Regev.
She slammed Palestinian President Abbas for not keeping his promise to release Gilad Shalit before the establishment of the Palestinian unity government.
"Now he cannot say that he has no control over Hamas," Livni told Ban. "He is sitting in the government with them. He is also responsible for the release."
Both sides, the Israeli and the Palestinian, had a good opportunity to present their arguments in the past few days. It is still unclear what will happen now, but it is clear that the UN chief brought along signs of change.
Ban estimated that within a breakthrough would take place within weeks or months. He reiterated that he was a true friend of peace and that he planned to finally bring peace to the Middle East.