The project, called The Hill, will be a foil to neighbouring Tower Bridge and the "Gherkin", with its inspiration drawn from natural forms. The structure, billed as a "green oasis" among the stonework and glass buildings along the Thames, is proposed for the Potters Fields area.
The plan was spearheaded by Arts Igniting Minds director Simon Elliot, with the shape of the building designed by Kathryn Findlay.
Findlay said: "Our design for The Hill will be a 'must see' experience in London, incorporating the need for green space in inner cities."
Plans for the gardens on the surface of the structure have been created by Olympic Park-appointed design firm BBUK.
BBUK director Jonnie Bell said he had been inspired by the wilds of Dartmoor in creating a concept for the landscape. He said: "We've been looking at very relaxed, shaggy wildflower planting. These are interesting and challenging gardens and we're very much looking at making it a public park."
Snaking pathways up the side of The Hill will provide access for pushchairs and wheelchairs, while more adventurous slides might link the paths. Bell said: "We were very keen that it shouldn't be too manicured and I think it will be quite low planting with a few wooded areas.
"It is aimed at everyone and the idea is that you could go and have a picnic up the hill. It's a much-loved local park at the moment so there's been a lot of public consultation."