People fleeing the Russian invasion of Ukraine have been arriving at the Polish border
Prime Minister Boris Johnson says the UK will "continue to bring maximum pressure to bear" on Russia as he prepares to fly to Poland and Estonia.
He said Vladimir Putin would "feel the consequences" for invading Ukraine.
Moscow has blamed comments by UK Foreign Secretary Liz Truss for the Russian president ordering its nuclear forces to be put on high alert.
The Russian army continues its advance on Kyiv, with satellite images spotting an armoured convoy about 40 miles long.
Air raid sirens were sounded again in the capital overnight, and there were also reports of fierce shelling in other cities including Chernihiv in the north.
It comes after fighting escalated on Monday, with missiles killing dozens of civilians in Kharkiv, Ukraine's second city. President Volodymyr Zelensky described the bombardment as a war crime.
Mr Johnson is due to meet the leaders of Estonia and Poland to discuss the response to the unfolding humanitarian crisis on Ukraine's border, as well as European security.
He will also speak with Nato Secretary General Jens Stoltenberg and meet British troops serving in Estonia - which shares a border with Russia.
Ahead of the trip, Mr Johnson said: "Alongside all our international allies, the UK will continue to bring maximum pressure to bear on Putin's regime to ensure he feels the consequences of his actions in Ukraine.
"We speak with one voice when we say, Putin must fail."
Boris Johnson at RAF Brize Norton in Oxfordshire on Saturday
How best to help Ukrainian refugees may be one subject Mr Johnson will discuss with his Polish counterpart on Tuesday. Poland says it has been trying to process more than 300,000 people who have fled from Ukraine.
The UK government is facing calls to make its visa rules more clear and to match the European Union's offer to accept Ukrainian refugees for up to three years, without asking them to apply for asylum.
Home Secretary Priti Patel said she would not waive the visa requirement for Ukrainians fleeing the conflict, but said the UK would extend visa rules to "immediate family members" to allow Ukrainians to "seek sanctuary" in Britain.