The farm, at the Wick of Gruting, would comprise 12 cages of 120m circumference arranged in two groups of six, the Shetland News website reported.
The equipment required would cost more than £2 million and the farm would create four full-time jobs.
It would be serviced from the existing Uyeasound pier in Unst and use a 200-tonne capacity SeaCap feed barge.
Smolts would be delivered via a wellboat and fish would be harvested dead haul and landed at Cullivoe in Yell.
No significant impact
An environmental impact assessment has already been carried out, which says that providing that certain conditions are adhered to, the farm is unlikely to have a significant environmental impact.
The Shetland News reported that in response to the planning application, Scottish Natural Heritage said that there are “interests of national and international importance on the site, but in our view these will not be adversely affected by the proposal”.
Shetland Shellfish Management Organisation said some concerns had been raised from fishermen about how exposed the site is and what would happen to shellfish and whitefish stocks should anything happen in storm conditions.
It had no objection to the application provided that trip ropes are moved from anchors after a few weeks and the site is adequately marked and lit for navigational safety.
Fetlar, known as the “Garden of Shetland”, has fewer than 100 inhabitants and is the summer home of one of Britain's rarest breeding birds, the red necked phalarope.
About 80 pairs of whimbrel (15% of the UK breeding population) also nest in Fetlar, and other bird life includes red-throated diver, golden plover, ringed plover, dunlin, eider duck, Arctic skua, great skua, Arctic tern and oystercatchers.