Matthew Gunner, who created the petition, argues that it would improve health services in the city, by making it easier to recruit and retain nurses.
At present, the expense of living in London means that the city's NHS suffers from a high staff turnover. Providing free travel would "mean that staff could afford a good quality of life whilst still working in the city."
He cites a similar arrangement currently offered to Metropolitan Police employees.
At time of writing, the petition has been signed by 397 people. It will remain open until next February.
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