Many residents of Kyiv and its outskirts packed up and departed for the countryside or an EU border as Russian air assaults continued over the night.
Traffic jams clogged roads in several towns, and some individuals walked into Poland and Hungary on foot.
The UN refugee chief said the humanitarian consequences will be “catastrophic.”
“We are very concerned about displacement – about people on the move,” Filippo Grandi, UN High Commissioner for Refugees, said in a BBC interview with Lyse Doucet in Kyiv.
“We believe that more than 100,000 individuals have already fled Ukraine in search of safety elsewhere in the nation,” he added, from Geneva.
Mr Grandi said that his organization had prepared to distribute supplies in Ukraine, but it was prevented from doing so during the fighting.
The European neighbors of Ukraine say they are prepared to welcome refugees from the country of 44 million people, which is one of Europe’s most populous.
Poland has pledged to prepare hospitals for the arrival of Ukrainian wounded and to establish welcome sites for refugees along its provincial boundaries.
Interior Minister Mariusz Kaminski said he was preparing for a “tsunami of refugees.”
“Poland is a secure nation for our people, it will also be a secure country for our neighbors,” he told Reuters.
According to Ukraine’s Interior Ministry, as many as 30,000 people have already fled from Ukraine to Moldova, which is located to the south.
“We anticipate considerably larger quantities,” Roland Schilling, the UN Refugee Agency‘s regional representative, said to the BBC.
“We are aware of long queues at border crossings. We’re working with other UN agencies to prepare our humanitarian response.”
The new Hungarian government has said it will send troops to the country’s southern borders “to protect Hungary’s border” and deal with any influx of asylum seekers.
Families are trekking across the border on foot with their suitcases, as hundreds of refugees have arrived at Hungary’s and Poland’s borders over the past few days.
Even so, the number of people living in Ukraine’s major cities is still very small, as many are finding it difficult to leave amidst the Russian-Ukrainian conflict.
At 5 a.m., in the north-eastern city of Kharviv, Ukraine, Victoria Vota heard what she called “exploding sounds.”
“My neighbors were ringing my doorbell to tell me it was time to leave the city,” she continued.
Despite mixed signals on whether or not it’s safe to stay put, “we’re unsure if it’s safe to evacuate,” she said. Later in the morning, some of the roads out of town came to a halt.