The number of refugees fleeing the war in Ukraine is expected to top two million in the next two days, the head of the UN refugee agency said Tuesday.

"I do think that we will pass the two million mark today or maybe at the latest tomorrow. So, it doesn't stop," Filippo Grandi, the UN High Commissioner for Refugees, told reporters in Oslo.

On Monday, the UNHCR put the number of refugees fleeing Ukraine at more than 1.7 million.

Grandi made his remarks at a press conference, after visiting Moldova, Poland and Romania, all of which have received refugees pouring across the border from Ukraine since Russia invaded the country on February 24.

He praised the "exemplary" welcome provided by these three countries, adding they seemed to be "coping" with the "natural spontaneous distribution."

Grandi stressed that the first waves of refugees were those with "some resources,"

"Many come by car, and especially they have connections. They can go where they have family, friends, communities," the commissioner said.

"It is possible that if the war continues... we will start seeing people that have no resources and no connections and that will be a more complex issue for European countries to manage going forward and there will need to be even more solidarity by everybody in Europe and beyond," he said.    

For comparison, Grandi said the Balkan wars in Bosnia and Kosovo saw "maybe two to three million people, but over a period of eight years." 

While other parts of the "world have seen this," Grandi added, "in Europe it's the first time since the Second World War."

On Monday, Grandi called it "the fastest growing refugee crisis in Europe since World War II," in a post to Twitter. 

After several failed attempts, Russia promised to open humanitarian corridors on Tuesday to allow civilians to flee the Ukrainian cities that have come under artillery fire.