The Mermaid was carrying South Korean tourists when it was in collision with a cruise ship and capsized, leaving 20 people dead and eight others missing.
Rescue efforts in the Hungarian capital have been hampered by high water levels, made worse by heavy rain, BBC reports.
Footage from the scene on Tuesday showed a floating crane slowly winching the boat to the surface.
Before the Mermaid is fully raised, divers will enter the wreck to search for any bodies still trapped inside, officials said.