The boat was struck by another vessel while moored in the Hungarian capital at about 10pm local time (8pm GMT), state news agency MTI reports.
According to MTI, there were 33 people on board when the incident happened, mainly South Korean tourists, as well as two crew.
Reuters reports 21 people, all South Korean, were still missing on Thursday morning (30 May).
The tourists were travelling with Korea’s Very Good Tour company.
The incident happened near the Hungarian parliament building, one of the city’s most popular tourist hotspots.
Gabor Csato, head of the Hungarian ambulance service, said rescue efforts had been hampered by the strong current on the Danube, exacerbated by recent heavy rain, The Guardian reports.
Hundreds of tourist boats of varying size sail sightseeing tours of the city every day, with many of the city’s iconic sights situated beside the river.