The ultimatum to the rebels of eastern Qalamoun was served to civilians from the area during a meeting with a Russian colonel and an officer from the Syrian Air Force Intelligence, rebel spokesman Said Saif of the Martyr Ahmad Abdo told Reuters.
The area is 40 km (25 miles) from Damascus and separate from eastern Ghouta - an area right next to the capital where government forces have routed rebels in recent weeks, forcing thousands to accept safe passage out towards the Turkish border.
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An opposition source said talks over the densely-populated town of Douma, the last rebel pocket of eastern Ghouta, had yet to be concluded, at odds with state media reports that the group holed up inside - Jaish al-Islam - had accepted a deal to leave.
President Bashar al-Assad, his military position secure thanks to Russia and Iran, is seeking to crush the last pockets of opposition to his rule near the main cities of western Syria.
The rebel enclave in eastern Qalamoun includes several towns and a barren expanse of mountainous territory.
"A clear message was sent to the Free Syrian Army (FSA) groups in the area: either reconciliation and disarmament - handing weapons to theSyrian government as the Russians describe it - or departing eastern Qalamoun," Saif said in separate comments to al-Hadath TV.
Saif said rebels had made a proposal under which they would withdraw from the towns into the mountains and civilians would stay, and Russia's response was being awaited. He said the aim was to avoid the "forced displacement" of people that had occurred in other areas recovered by the Syrian government.
The Syrian military could not immediately be reached for comment.
Saif told Reuters the civilians who met the Russian and Syrian military officers had expressed concern that a departure of rebel fighters would leave them vulnerable to attack by jihadists from Nusra Front or Islamic State.
The Syrian war last month entered its eighth year having killed hundreds of thousands of people and forced 11 million from their homes, including nearly 6 million who have fled abroad in one of the worst refugee crises of modern times.
While Assad now controls the single largest chunk of Syria, it may prove difficult for him to regain much more territory without colliding with the interests of foreign states, notably Turkey and the United States which have forces in the country.
Russian-backed government forces have recovered nearly all of eastern Ghouta in a ferocious assault that began in February. The Syrian Observatory for Human Rights says bombardment of the rebel enclave killed more than 1,600 civilians in Ghouta.
It marks Assad's most significant victory over the rebellion against his rule since rebels were driven from eastern Aleppo in 2016.
State media has said Jaish al-Islam had accepted a deal giving its fighters safe passage to towns at the border with Turkey that are located in a buffer zone controlled by the Turkish military and allied Syrian rebel groups.
But while Russia's defence ministry said 2,000 had left since April 1, the opposition source familiar with the negotiations said no final deal had been reached with Jaish al-Islam.
The group, which is estimated to have many thousands of fighters, has previously insisted it will not leave Douma or accept "forced displacement" to another part of Syria.
The source said that while Jaish al-Islam wanted to stay in the town, it was not seeking its own "independent canton". People in Douma wanted a reconciliation deal with the state that keeps out its feared security services, the source added.
A military source told Reuters on Monday that some elements of Jaish al-Islam were still rejecting a deal and that military force would be used if they refused to strike one.
The source said on Tuesday the government had set a deadline for militants to leave Douma, without saying how long.
"The Syrian government has taken a decision to clear militants from the areas," the source told Reuters.