A total of 4,330 migrants entered the country on Saturday, outstripping the previous one-day record of 3,601 set on Thursday.
Hungarian authorities said they feared a new surge of arrivals on Sunday.
Almost all the migrants are heading towards Germany via Austria. Austrian police said some 6,700 crossed the Austria-Hungarian border at Nickelsdorf on Saturday before heading north toward Germany. They predicted similar numbers for Sunday.
Hungary is trying to seal its border with Serbia to stop thousands of migrants and refugees traveling up from Greece and the Balkans.
On Tuesday, a new law is due to come into effect which could see migrants jailed for trying to cross a 175-kilometre (120-mile) razor-wire fence it is building along the border.
The UN refugee agency has warned that criminalising illegal border crossings could violate the UN Convention on Refugees if it involved asylum seekers.
