There are also hopes it will clear the way to support its tenants by maintaining affordable rents and building new homes in the community.
The proposal was presented to Minister for Local Government and Planning, Derek Mackay MSP, by Shetland Islands Council's Political Leader Gary Robinson.
Mr Robinson said: "We have taken the initiative to try to break the deadlock on Shetland's historic housing debt problem and secure a better future for our tenants in terms of affordable and available housing.
"What we are proposing is to ask the UK Government for £10 million which we would match, taking the debt down to £20 million and allowing rents to be kept at an affordable level.
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"We are also asking the Scottish Government for £10 million of housing grant for the period 2016-18, which when matched by a similar sum from us in terms of land value and funding for housing development would pave the way for new affordable homes to be built in Shetland.
"We are pleased that the Scottish Government has taken our proposal away to consider in more detail and we hope that they can encourage the UK Government to sit down and discuss the matter.
"In terms of next steps, Minister for Housing and Welfare, Margaret Burgess MSP, is coming to Shetland on 7 August. We hope she will be in a position to give us feedback on the proposal when she meets with us.
"The position we have put forward is, in our minds, one that could settle the housing debt debate once and for all."
SIC's Executive Manager for Housing, Anita Jamieson, added: "A solution to the debt issue is long overdue and is crucial to protecting our tenants, by ensuring that we can provide a quality service that meets their needs and expectations at an affordable rent. The views of our tenants were well captured by the Shetland Tenants Forum's recent survey.
"We also need to find ways of investing in new housing supply to meet the growing needs of those on the waiting list. The proposal put forward is designed to find a sustainable way forward."
(JP/MH)
