The three families' lawyers have filed claims on their behalf.
Some city legislators say two Sri Lankan nationals have been targeted by police from their own country who have travelled to Hong Kong.
The refugee families fed and housed Mr Snowden for two weeks when he fled the US after leaking thousands of files.
They are still living in poverty in Hong Kong and unable to work after providing a refuge for the American in July 2013.
Their lawyers say it is "a matter of life and death" if they remain in the city.
Mr Snowden - who now lives in Russia - tweeted on Thursday that he hoped Canada would grant their request.