The boat was moored near the Hungarian parliament building and reportedly was hit by another tourist vessel, then turned over on the river, which has been flooding with strong currents while a rainstorm also enveloped Budapest.
Reuters could not immediately verify the information about the victims.
MTI said the boat carried 34 people, including crew, mostly Asian tourists. State television said there were 32 tourists and two crew on board.
A police spokeswoman and the disaster units spokesman could not immediately comment.
The news web site Index.hu cited the boat’s owner as reporting that the boat, called Mermaid, had sunk.
Rescue efforts were under way with more than a dozen vehicles visible several hundred meters downstream from the site of the accident.
Eighteen people were accounted for, with three dead and sixteen missing, state media reported, citing authorities.
At least one tourist had to be revived, the National Ambulance Service told state news agency MTI.
Television footage shows the Danube bank closed off by police on the Pest bank, directly across from the World Heritage site of Buda Castle.
Police ordered all shipping to stop on the Danube at central Budapest, it said in a statement.