The stakes couldn't be higher. An agreement between Putin — who has backed the Syrian government of Bashar Assad in Syria's multi-faceted war, and Turkish President Recep Tayyip Erdogan could definitively end the fighting along the Turkey-Syria border.
But failure to reach a deal could set the stage for a wider conflict that pulls in Turkey, Russia, Syrian government forces, Kurdish militias and Turkey-backed Syrian fighters.
Tuesday's meeting in Sochi, Russia, will come just hours before the clock runs out on a cease-fire agreed between Turkey and the United States. Under that agreement, Turkish troops and their allied Syrian fighters stopped fighting for five days, on condition the Kurdish fighters holding the area withdraw. The agreement has largely met Turkey's aims, but left the Kurds feeling betrayed by Washington, which is currently pulling U.S. troops out of the area.
Any deal Russia reaches with Turkey will have to address what happens to Syrian government forces and Russians that have moved into limited areas in the border region, and the fact that much of northeastern Syria has majority Kurdish populations.
Erdogan launched the military offensive into Syria on Oct. 9 with two stated aims: Clearing the area of Kurdish fighters and resettling about 2 million of the 3.6 million Syrian refugees currently living in Turkey in new towns and cities Turkey will build there.
Turkey describes the Kurdish fighters as terrorists, seeing them as little more than an extension of a separatist Kurdish group which has been fighting inside Turkey on and off since the 1980s.
Erdogan has long complained about the presence of the Kurdish fighters on Turkey's borders, and Ankara was aghast at the creation of what was essentially an autonomous Kurdish region in northern Syria.
Erdogan has said the Turkish-patrolled "safe zone" in Syria into which the refugees would be resettled would extend about 30 kilometers into Syria and running 444 kilometers (275 miles) along the Syrian-Turkish border.
Since much of the border area is predominantly Kurdish, this has raised concerns of an attempt to alter the region's demographics.
Turkish forces and Turkish-backed Syrian opposition fighters moved in days after U.S. President Donald Trump announced he was withdrawing American forces. Trump's decision was widely seen as an abandonment of Kurdish fighters who had died in the thousands working with the U.S. to combat the Islamic State group.
Although Turkish officials say the ceasefire agreement specifically covers a roughly 120-kilometer (75 mile) stretch between the Syrian border towns of Tal Abyad and Ras al-Ayn, and extends 30 kilometers into Syria, Erdogan has made clear he wants Turkish military presence along the full stretch of the border from the Euphrates River to Syria's border with Iraq.
But there's an added complication.
Abandoned by their former allies, the Kurds turned to Assad for protection from Turkey.
This has led to the presence of Syrian government troops in limited areas on the Turkish-Syrian border, namely in the Kurdish-majority town of Kobani, once a flashpoint in the fight against IS.
Meanwhile Russia, which often has advisers backing Syrian forces, has raised a Russian flag east of the Turkish-controlled town of Jarablus.
Russian Foreign Minister Sergey Lavrov said Monday that Syria's Kurds should be incorporated into the country's legal structure so they do not pose a threat to Turkey.
Russia's "goal is to create a situation where all Kurdish organizations in Syria are woven into the country's legal framework and constitution, so that there are no illegal armed units in Syria," he said.
Putin's foreign affairs adviser Yuri Ushakov said long-term stability in the region can only be achieved by restoring Syria's territorial integrity and respecting all ethnic groups. But he refused to answer questions on whether Russia expects Turkey to withdraw its forces from Syria or spell out any other details of the Kremlin's stance.
For its part, the Syrian government has not commented on the upcoming meeting. Damascus has said next to nothing about the agreement reached through the Russians to allow Syrian troops to deploy in positions along the border and other spots as U.S. troops withdraw.
On Friday, Assad received Putin's envoy for Syria, Alexander Lavrentiyev. State media quoted Assad as saying the focus should now be on stopping the Turkish offensive and the withdrawal of Turkish and American forces "since they are occupation forces."
A senior Kurdish official, Redur Khalil, told The Associated Press they want the meeting to come up with a decision to offer international protection to the area taken over by Turkey. He said they want "a permanent cease-fire" that would guarantee Turkey won't expand its presence in Syria.
Khalil said the protection would be for civilians who want to return home and are afraid of Turkey's plans for demographic changes, or revenge attacks from Turkey-backed Syrian fighters . While those fighters include some Islamic extremists and past members of some Syrian rebel factions, many are Arab and Turkmen fighters from northern and eastern Syria who have an ax to grind against the Kurds and a reputation for violence and looting.
Most Kurdish civilians have fled the town of Ras al-Ayn, fearing killings or repression by those fighters. Any still in the territory the Kurdish fighters are leaving are likely to follow suit.
Both Kurdish and Turkish officials have said Kurdish forces are indeed withdrawing from the area in question, although each side has accused the other of violating the cease-fire on several occasions.
Turkish Foreign Minister Mevlut Cavusoglu warned on Monday that Turkey will resume its offensive if the Kurdish fighters don't vacate the area by the end of the cease-fire agreement on Tuesday evening.
"If they don't withdraw, our operation will restart," Cavusoglu said.
Suzan Fraser in Ankara, Sarah El Deeb in Beirut and Vladimir Isachenkov and Jim Heinz in Moscow contributed.