The winning team, which will be commissioned to raise the level of the Victorian gardens, is scheduled to be selected in November.
The contenders are Gustafson Porter, West 8, Olin Studio, Vladimir Djurovic Landscape Architecture (Beirut), Ian White and Aecom. Designs include bridges, curved glass and auditoria.
Consultant and competition organiser Malcolm Reading said: "The finalists have a record in designing spaces that intrigue, excite and refresh people."
Locals can comment on the entries at the nearby Academy until 2 November.