Trusts across England have been told to prepare for the programme to begin in early December, the Mail on Sunday reported.
However the timeline has not yet been confirmed as none of the vaccines currently being developed have gained clinical approval.
The newspaper quoted a memo to staff from Glen Burley, chief executive of George Eliot Hospital NHS Trust in Warwickshire.
He wrote: 'Our Trust, alongside NHS organisations nationally, has been told to be prepared to start a Covid-19 staff vaccine programme in early December.
“The latest intelligence states a coronavirus vaccine should be available this year with NHS staff prioritised prior to Christmas.”
Mr Burley also urged staff to get a flu shot by the end of November to qualify for a Covid-19 vaccine, which he said was "expected to be given in two doses, 28 days apart".
David Eltringham, managing director at the Trust, told the Mail: “We don't have a definite date for delivery of the vaccine, but we are making ready to deploy the vaccine from the beginning of December.”
The newspaper also reported that Diane Wake, chief executive of the Dudley Group NHS Trust, was "hoping for a Covid-19 vaccine to be available to healthcare providers some time in December". She is said to have told a board meeting: "It has not been confirmed yet but I'm hoping to be able to offer a Covid-19 vaccine to our staff."
NHS England said it did not have any further comment beyond the quotes from the Trust bosses.
Experts remain hopeful that a vaccine could be approved for use by the end of the year - albeit in limited numbers.
Patrick Vallance, the UK government’s chief scientific adviser, said earlier this week: "We may get a few doses this side of Christmas, but I think we should more realistically be looking at spring, and of course there are no guarantees."
The government has already ordered at least 380 million doses from a combination of manufacturers, equivalent to almost 6 shots for everybody in the UK.
One of the leading contenders for a vaccine is being developed by Oxford University researchers alongside the drug company AstraZeneca.
The Department of Health and Social Care has been approached for comment.