The authority is looking ahead to how a mass vaccination programme could be delivered in the county in the coming months.
And Staffordshire is one of 66 areas chosen to roll out of rapid turnaround lateral tests, which can give results in under an hour without the need for laboratory processing, it was announced last week.
But at Tuesday's Corporate Review Committee members heard the council was facing challenges in delivering the tests to residents.
Council leader Alan White said: "We thought lateral flow testing would be coming in this week but it’s not going to be this week now because of various logistical issues.
"We have submitted a number of requests for military assistance. We have 11 Signals Brigade based in Stafford and the Defence Medical Services based in Whittington.
"Those requests have not come back with a positive response. We have also flagged up to the Covid-19 representative from the Cabinet Office who is also a military officer to say we have been making these requests and could you do what you can to help.
"Whilst those requests have been made they have so far fallen on deaf ears but we will continue to make them and hopefully we will get some support."
Dr Richard Harling, the council’s director of health and care, said: “These are new near patient tests which provide an opportunity to identify cases and their contacts more quickly.
"If we can get it rolled out at scale it is likely to have some impact on the spread of infection, although we do need to manage expectations as the virus is not going to disappear suddenly overnight because of an introduction of these tests.
“We expect to have them available at scale – the Government has indicated a supply of up to 10 per cent of the population a week. The main constraint is they require a trained operator and there is no associated funding for people to deploy the tests.
“We are going to have to think imaginatively about how we get a workforce to be able to provide the tests. We have had a conversation with the district and borough councils.
“We envisage using these tests in a range of settings; for example making them available to public sector partners – in which we would need their own occupational health departments to provide the trained operators; making them available to large employers – again we would look to them to provide the workforce; and then to look to deploy them in those communities that have the highest rates of infection.
"In that scenario I think we would need to deploy volunteers and we have initiated conversations with the district and borough councils about how we mobilise a volunteer workforce to provide that capacity."
Dr Harling said the NHS would be taking the lead on administering vaccinations – but said the "main constraint" would be at the speed at which supplies of the vaccination come through the pipeline.