If successful, Visionary Crazy Golf will take over the area for the first week of the festival, which runs from 17-25 September.
The course will feature nine unusual holes designed by emerging studios and big names, all themed around 'cities of the future'.
Architect Zaha Hadid, who died earlier this year, devised a curved, split-level construction that follows the shadow of Nelson's Column, meanwhile designer Paul Smith, the project's ambassador, has created a brightly coloured run down the steps of the National Gallery. It will also feature a 'clubhouse' that is complete with turf roof and putter-shaped columns.
Other ideas include a series of pneumatic tubes by Tom Dixon that are designed to evoke future modes of transport, and a giant pigeon by Ordinary Architecture that has been sliced in half to show golf balls moving through its digestive system.
Permission for the installation has already been granted by the Mayor’s office, however Visionary Crazy Golf will need to hit its £120,000 goal in order to bring the idea to life.
Backers can receive gifts including a tote bag from designer Camille Walala and socks from Paul Smith. Those who pledge £5,000 or more will be rewarded with a tour of Paul Smith's studio and have their name featured in the clubhouse. They can also opt to play a round of golf with the designer at the opening on 16 September.
This year is the The London Design Festival's thirteenth in the capital, and it will encompass more than 400 different events.
A giant Monopoly board was recently positioned in Trafalgar Square, to celebrate the launch of the first London Games Festival.