If all goes well, the wreck will be placed on a barge near the crane, from where it will be taken over by Budapest police investigating the collision.
Before the Mermaid is fully raised, divers will enter the wreck to search for any bodies still trapped inside, officials said. Pumps were used to remove water from the hull and the rest of the tourist boat was slowly brought to the surface.
Hungarian rescue officials said Monday they have almost completed their preparations to start the hard process of raising a sunken tour boat from the floor of the Danube River.
Early on in the operation, police were seen by an AFP photographer at the site removing four bodies from the vessel.
Nineteen South Korean tourists and a Hungarian crewman were previously confirmed to have died, with eight people still listed as missing.
Mermaid was a 27-metre (89-feet) double-decker river cruise boat with a 150-horsepower engine and had a capacity to hold up to 60 people.
The raising of the sunken Danube River tour boat is underway in Hungary, with parts of the boat already visible on the river's surface.
A floating crane raised the boat from waters beneath the Margit bridge in downtown Budapest, live TV footage showed.
State news agency MTI reported that a total of four victims had been found on Tuesday and recovered by divers. Salvage crews saluted in respect as each body was removed. Seven people were rescued after the nigh-time collision in heavy rain.
Recovery efforts had been hampered by high springtime water levels, strong currents and near-zero visibility under the waterline.
The captain of the river cruise ship, the Viking Sigyn, has been arrested on suspicion of "endangering waterborne traffic resulting in multiple deaths".
Police on Sunday carried out another inspection of the Viking Sigyn, which left Budapest less than 48 hours after the collision with the Hableany but was back in Hungary on a scheduled trip and docked at the town of Visegrad, north of Budapest.