The island's MP, Ian Blackford, said 28 out of 34 residents and 26 out of 52 staff had tested positive for Covid-19 at Home Farm care home in Portree.
NHS Highland and the local council say there are no signs the virus has spread further into the community on the island.
Mr Blackford posted a series of tweets about the situation on Sunday.
He said: "The sad news is that 28 of the 34 residents have tested positive for Covid-19 as have 26 of the 52 staff.
"My thoughts are with all of those who are infected and I know the entire community of Skye will want to rally round and offer support to all of those affected.
"Testing and contact tracing will be important as we all seek to see the control and eradication of this outbreak on Skye and all of us on the island have our part to play in this."
He said the Army would establish a mobile testing facility nearby on Monday afternoon.
He added: "Having such an outbreak as we have on Skye is shocking and worrying.
"By supporting the NHS and our care staff, by protecting all our frontline workers, and by taking all necessary hygiene steps we can contain the spread of this virus and do our bit to save lives."
Local MSP Kate Forbes said: "The NHS, with Highland Council, have moved quickly to contact all staff and households to ensure they self-isolate. Staff and residents have been tested.
"There is already testing facilities in Portree, and that will be complemented by a mobile testing unit in Broadford."
Home Farm is operated by HC-One and offers nursing and nursing memory care for those with dementia.
The first cases were reported a few days ago and extra staff have been brought in to help care for the elderly residents.
A joint statement from NHS Highland and Highland Council said there had been a "significant" outbreak at the home.
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