Seven people so far are confirmed dead and seven others were rescued.
Foreign Minister Peter Szijjarto and his South Korean counterpart, Kang Kyung-wha, visited the site of Wednesday's accident and met with rescue and search officials before holding talks.
"We have to prepare for a protracted search," Szijjarto said he was told by Budapest river police, adding that authorities were also expecting help from Serbia, where the Danube flows after leaving Hungary.
"It is not unimaginable that we will need to depend on them in the coming days."
Divers in full equipment went into Danube but were unable to inspect the wreckage of the boat due to the extremely murky waters and heavy currents from recent rains. Officials said it could take days to recover the 70-year-old boat, which was built in the former Soviet Union.
Just hours after Wednesday night's collision, the body of one victim was found nearly 12km downstream from the site of the collision, near the Hungarian capital's neo-Gothic parliament building.
"The wreck is located more than six metres (20 feet) deep and the water level continues to rise because of the expected rainfall," Szijjarto added.
The Interior Ministry said some 60 experts were experimenting with theoretical models and test dives to find a "unique" method to salvage the sunken tour boat or rescue any bodies trapped inside. It said the adverse conditions don't allow for the use of any traditional methods.
Hungarian police have detained and questioned the captain of the Viking Sigyn river cruise ship that collided with a sightseeing boat. Szijjarto said the 64-year-old Ukrainian captain protested being considered a suspect. Identified only as Yuriy C., he is suspected of endangering water transport leading to a deadly mass accident.
The Viking Sigyn has been allowed to sail on to Germany but Kang said the ship's owner "has promised to fully cooperate with the investigation."
"If the investigation finds the ship's owner responsible, there will be a thorough legal response," she added.
The ministers agreed to search for every missing person.
"We shared our firm resolve ... to search for survivors to the last person," Kang said.
A South Korean group on a package tour of Europe — including 30 tourists, two guides and a photographer— were on an hour-long sightseeing tour of Budapest when their boat collided with a Viking cruise ship during a downpour Wednesday evening.
Nineteen South Koreans and two Hungarian crew members — the captain and his assistant — remain missing.
A crew member from another tour boat who helped rescue two South Korean women, said one of them was in shock.
"I announced 'Man overboard' on the radio in Hungarian and German and we started the rescue," Norbert Magyar told broadcaster RTL Klub. "We threw out two life preservers which were caught by two Korean ladies and with the help of our colleagues we lifted them out of the water. The passengers helped too, because ... the two women's clothes absorbed a lot of water and it would have been very difficult to lift them out."
The boat's sinking has touched a nerve in South Korea, where many are still traumatized over a 2014 ferry sinking that killed more than 300 people, mostly students.
Survivors say they were lucky to grab onto a drifting lifeboat Wednesday evening while looking in horror as others around them struggled in the dark, rainy waters, shouting for help.
Their small sightseeing boat had almost finished its tour and was almost at a stop when the larger cruise ship hit it under a bridge near the riverbank parliament building. They said about 20 people were on the deck taking photographs or preparing to disembark. The others were in the cabin.
"I saw that big cruise ship coming closer to us but I had never imagined it would ram our boat," a 31-year-old South Korean surnamed Jeong told the Yonhap news agency.
Jeong said she and others on the deck were thrown into the cold Danube by the impact of the collision, while police said it took only seven seconds for the tourist boat to overturn and sink. Jeong said she grabbed a lifeboat drifting by and managed to throw a rope to another tourist surnamed Yoon.
"Our boat was turned over in an instant and began sinking," Yoon, 32, told Yonhap. "All those on the deck fell into water. I think those staying in the cabin on the first floor couldn't probably get out of the ship quickly."
The South Korean government said no one was wearing a life jacket.
Video released by Hungarian police showed the sightseeing boat, identified as the Hableany (Mermaid), traveling closely side-by-side and in the same direction as the German-built Viking cruise ship as they approached the Margit Bridge.
The Hableany then appears to steer slightly to its left, into the path of the 135-meter (443-feet) long cruise ship, which continued to sail on at the same speed. The two collided and the sightseeing boat then tipped over on its side between the bridge's two supports.
Police Col. Adrian Pal said it was unclear what caused the Hableany to steer into the path of the Viking.
