media captionMany of the migrants are now living in a camp in Verebiejai
Lithuania's parliament has voted to build a fence on the Belarus border to stop Iraqis, Afghans and other non-EU migrants entering.
More than 4,000 migrants have entered EU member Lithuania illegally from Belarus this year – a record number.
Lithuanian officials accuse Belarus of flying in migrants who then head to the border – a claim Belarus denies.
Lithuania's help for Belarusian opposition activists has angered Belarus President Alexander Lukashenko.
Relations deteriorated further in June this year after Belarus forced a Ryanair flight heading to Lithuania to divert to Minsk and arrested a Belarusian opposition journalist and his girlfriend on board. The EU strengthened sanctions a month later.
Since then there has been a surge of non-EU migrants crossing from Belarus into Lithuania, Latvia and Poland, putting pressure on the EU to take action.
Latvia has declared a state of emergency on its border with Belarus, where it plans to beef up patrols. The measure authorises border guards to return illegal migrants by force.
Mr Lukashenko, however, vowed on Monday that Belarus would not accept migrants back from its Baltic neighbours. His disputed re-election a year ago triggered mass protests, violently broken up by his security forces.
Lithuania aims to build a 4m (13ft) metal fence topped with razor wire on the border with Belarus. It is expected to cost about €152m (£128m; $178m).
"Without this physical barrier, it is impossible to protect our borders, it is very clear," Interior Minister Agne Bilotaite told Reuters news agency.
But on Tuesday a spokesman for the EU said the bloc "does not finance fences or barriers". So far, the EU has offered help in the form of border guards and supplies instead.
The Lithuanian government has also announced a plan to offer migrants who agree to return to their country of origin €300 each, national broadcaster LRT reported.
A harsher and more controversial policy of pushing migrants back from the border began last week.
Many come from Iraq, according to the Lithuanian Border Guard, which has been detaining migrants at several temporary detention centres.
image sourceEPAimage captionLithuania has accused Belarus of flying in foreign migrants and allowing them to go to the border
The increasingly fraught situation on the border was acknowledged by President Lukashenko at a marathon eight-hour TV news conference in Belarus on Monday.
In power since 1994, Mr Lukashenko accused Lithuania of deliberately provoking his government into a reaction at the border.
"If they don't calm down at the borders, they will get an all-out sock on the jaw," Mr Lukashenko said.
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Belarus: The basics
- Where is Belarus? It has its ally Russia to the east and Ukraine to the south. To the north and west lie EU and Nato members Latvia, Lithuania and Poland.
- Why does it matter? Like Ukraine, this nation of 9.5 million is caught in rivalry between the West and Russia. President Alexander Lukashenko has been nicknamed "Europe's last dictator" – he has been in power for 27 years.
- What's going on there? There is a huge opposition movement demanding new, democratic leadership and economic reform. The opposition movement and Western governments say Mr Lukashenko rigged the 9 August election. Officially he won by a landslide. A huge police crackdown has curbed street protests and sent opposition leaders to prison or into exile.