Who prevails could have major ramifications for Israel's prospects of peace with the Palestinians.
Under pressure, Olmert has been sending mixed messages. Like the doves in the centrist Kadima Party, he says he believes a moderate Palestinian Authority leadership led by P.A. President Mahmoud Abbas represents a unique window of opportunity for peacemaking that should be explored fully.
But like the hawks, Olmert says a final-status agreement will take at least 20 to 30 years.
The prime minister is also caught between opposing pressures from the international community and his party's hard-liners.
The international community, led by the United States, wants Israel to reach agreement in principle with the Palestinians on the core issues of the conflict: final borders, the status of Jerusalem and the question of the right of return for Palestinian refugees.
The United States wants this to happen before the November parley, to which it has invited Syria and Saudi Arabia in an attempt to expand the summit's scope. As these Arab states have been reticent to open diplomatic channels with Israel amid Israeli-Palestinian fighting, progress on the Israeli-Palestinian track is essential to the success of broader Arab-Israeli peace and November's summit.
Kadima's hawks, however, are warning Olmert not to go too far, otherwise his government could fall.
The leading spokesman for Kadima's doves is Olmert's close confidant Haim Ramon, the deputy prime minister and former Labor Party man who has drafted a far-reaching joint agreement of principles with the Palestinians on the core issues of the conflict.
The hawks' ideologue is Knesset member Otniel Schneller, a West Bank settler from Ma'aleh Michmash, near Ramallah, who has drafted a detailed counterproposal that envisions a much slower peacemaking process that could last decades.
In early September, in what was widely believed to be a trial balloon for Olmert, Ramon leaked elements of his draft plan to Israel's daily Yediot Achronot. On the matter of Jerusalem he proposed a simple split: In a final peace deal with the Palestinians, Jewish neighborhoods would go to Israel, Arab neighborhoods to the future Palestinian state.
Though this is not a new idea -- it had been raised by then-Prime Minister Ehud Barak in the Camp David negotiations with Yasser Arafat seven years ago -- it infuriated Kadima's hard-liners, many of whom had come to Kadima from Likud.
At a key meeting of Kadima's leadership last week, Transportation Minister Shaul Mofaz argued that it was premature to talk about final-status issues with the Palestinians.
"I urge all those who have lost patience to learn from the lessons of the past and to consider how all the wild rushes towards final settlement have ended," declared Mofaz, a former defense minister under Likud.
Mofaz's attack on the current Israeli-Palestinian peace process encapsulated the divisions in the ruling Kadima Party.
The doves believe the West Bank Palestinians now have a moderate leadership committed to peace, and therefore a unique opportunity exists for accommodation that should not be missed. The hard-liners counter that this belief is a dangerous illusion that will lead only to more bloodshed and a Hamas takeover of the West Bank -- as happened in June in the Gaza Strip -- as soon as Israel withdraws from the area.
Schneller in particular warns against any attempt to force final-status issues. In a detailed peace plan he calls "The Complete Vision," Schneller argues that the reality on the ground must change -- along with Palestinian and Israeli hearts and minds -- before peace can be attempted.
In Schneller's view, past peacemaking attempts failed because Israelis and Palestinians were not truly reconciled and because Israelis were divided about how to deal with the Palestinians. Israelis and Palestinians first must establish "good neighborly relations," and Israelis must reach a wide national consensus on the general terms of a peace package, before negotiations can begin.
Once that is achieved -- no mean feat -- Schneller proposes five stages of negotiation, each of which could take several years.
Schneller says he has been "greatly encouraged" by responses to his plan from Olmert and other Cabinet ministers, and that Ramon's peace plan has no chance of being accepted in the party.
"Sixty percent of Kadima voters are ex-Likudniks, 30 percent Labor and 10 percent Shinui," he said. "They do not identify with Ramon's radical left-wing ideas."
Olmert's difficulties on the Palestinian front don't end with Kadima's hawks. His right-wing coalition partners, Shas and Yisrael Beiteinu, both have warned they will bolt the coalition if he makes concessions on any of the core issues.
There is another major snag: The Palestinians don't accept the Ramon formula either.
In negotiations so far, major differences have emerged between the two sides on form and substance regarding final-status issues.
On form, Abbas wants a detailed, binding declaration of principles while Olmert wants a more vague joint statement.
On substance, Ramon talks about Israel's West Bank security barrier as the basis for a final border between two states, while the Palestinian Authority talks about the 1967 Green Line. The Palestinians want compensation for any land annexed by Israel on a one-for-one basis; Ramon says that in land swaps quality, not just quantity, must be taken into account.
The Palestinians want any agreement to include a safe passage between the West Bank and Gaza; Ramon says that has no place in a general agreement of principles.
Perhaps most important, the Palestinians want a timetable for implementation -- something the Israelis are loath to give.
For now, Olmert is keeping his options open. On Monday he told the Knesset's Foreign Affairs and Defense Committee that in a peace deal Israel would withdraw from large tracts of the West Bank, but that was unlikely to happen for another 20 or 30 years.
Schneller says Olmert is too smart a politician to allow Kadima to collapse over an illusory peace process.
Whether or not Olmert will be able to continue straddling the line between his party's hawks and doves come November, when the eyes of Washington and the world will be on the Israeli-Palestinian track, remains to be seen.
Leslie Susser is the diplomatic correspondent of the Jerusalem Report.