“Our information is that special trains are taking migrants from Röszke (train) station direct without stopping to the Austrian border,” Erno Simon, UNHCR Regional Representative for Central Europe, told AFP.
He said this was “approximately a four-hour journey, yesterday (Sunday) three such trains left carrying at least 2,000 people. During the night our colleagues saw police waking people up at the border collection point.”
There was no immediate comment from the Hungarian government about the development, which matches comments made by migrants to an AFP correspondent at the flashpoint town of Röszke near the Serbian border.
Hungary is supposed to give asylum seekers medium-term housing in refugee camps, of which Hungary has several, although they are reportedly full.
EU member Hungary has become a frontline state in Europe's refugee crisis, with more than 180,000 people travelling up from Greece through the western Balkans and entering the country this year.
On Sunday, police recorded a record 5,809 people entering Hungary, smashing the previous day's record of 4,330.
The sharp increase came ahead of harsh new Hungarian laws coming into force on Tuesday under which people entering the EU country illegally can be jailed for up to three years.
One unconfirmed report also suggests that Hungary plans to declare a state of emergency, as part of a plan designed to prevent asylum seekers from entering the country.
In addition to the new laws, Hungary is also building a controversial four-metre high (13-feet) fence all along its 175-kilometre (110-mile) border with Serbia.
The migrants, mostly Syrians, Afghans and Pakistanis, seek to travel onwards to Austria and then western Europe, particularly Germany — which has relaxed asylum rules for Syrians — and Sweden.
On Sunday, however, Germany reimposed border controls and trains from Austria were suspended — although they resumed on Monday morning — leaving thousands arriving from Hungary effectively stranded.
Austria's government responded on Monday with plans to deploy some 2,200 military personnel, primarily for humanitarian support but also to strengthen border checks.
By around midday there were more than 7,000 migrants at Nickelsdorf, the main crossing point into Austria, police said. Around 5,000 others spent the night in Vienna.