Escape routes to Russia and Belarus, its close ally, were earlier called "completely immoral" by a spokesperson for Ukrainian President Volodymyr Zelenskiy.
A fourth round of talks is due soon, Russian negotiator Leonid Slutsky told Russian state television.
"Our president is not scared of anything, including a direct meeting with Putin," said Ukraine's foreign minister Dmytro Kuleba late on Monday.
Kremlin spokesman Dmitry Peskov told Reuters Moscow would halt operations if Ukraine ceased fighting, amended its constitution to declare neutrality, and recognised Russia's annexation of Crimea and the independence of regions held by Russian-backed separatists.
French President Emmanuel Macron, who spoke with Putin multiple times last month in the run-up to the invasion, said he saw no impeding breakthrough.
"I don't think that in the coming days and weeks, there will be a real negotiated solution", he said.
FAILED EVACUATIONS
The general staff of Ukraine's armed forces said Russian forces were "beginning to accumulate resources for the storming of Kyiv", a city of more than 3 million, after days of slow progress in their main advance south from Belarus.
Outside the capital, attacks continued.
A Russian strike on a bread factory killed 13 in the town of Makariv in the Kyiv region, Ukrainian officials said. Reuters could not verify the details. Russia denies targeting civilians.
In the encircled southern port city of Mariupol, hundreds of thousands of people remained trapped without food and water under regular bombardments.
Deputy mayor Sergei Orlov told CNN that authorities were ready to evacuate 6,000 people on Saturday but Russians had bombed buses that were to transport them. Moscow has accused the Ukrainians of blocking the planned evacuations.
In the eastern city of Kharkiv, police said the total death toll from the Russian bombardment was 143 since the start of the invasion. It was not possible to verify the toll.
In Irpin, people picked their way over the twisted ruins of a large bridge.
"It's like a disaster," a young woman leaving with her children told Reuters.
Ukraine said on Monday its forces had retaken control of the town of Chuhuiv in the northeast after heavy fighting and of the strategic Mykolayiv airport in the south. Neither could immediately be verified.
In a humanitarian update, the United Nations described one psychiatric hospital 60 km (40 miles) from Kyiv running out of water and medicine with 670 people trapped inside, including bedridden patients with severe needs.
US congressional negotiators on Monday were nearing a deal on a bill to provide Ukraine with billions of dollars in emergency aid.
A senior US defence official said Putin had now deployed into Ukraine nearly 100 per cent of the more than 150,000 forces that he had pre-staged outside the country before the invasion.
Moscow has acknowledged nearly 500 deaths among its soldiers, but Western countries say the true number is much higher and Ukraine says it is many thousands.
Death tolls cannot be verified, but footage filmed across Ukraine shows burnt-out wreckage of Russian tanks and armour, and parts of Ukrainian cities reduced to rubble by Russian strikes.