The health board has had to request extra supplies after being unable to cope with demand for the vaccines.
The suspension applies to all workers apart from those in the highest priority areas and staff at care homes.
NHS Ayrshire and Arran blamed nationwide supply problems for the issue, but union leaders said no other health board had reported similar problems.
It comes after pharmacies across Scotland were forced to turn customers away from private flu vaccination appointments last week due to stock shortages.
In a statement, NHS Ayrshire and Arran said: "We are working closely with National Procurement to understand when additional supplies may be available in order to plan the next phase of the programme which will run until February 2021."
Unison spokesman Gordon McKay told the BBC: "If there is a national supply issue, then that is a major concern and the Scottish government need to address that.
"If, however, the issue is solely in NHS Ayrshire & Arran, the chief executive needs to be open and transparent with staff."
He added: “Both the Scottish government and senior clinicians have said, this year more than ever, it is important for staff to get the flu vaccine.
"Not just for the protection of staff or the protection of their families and communities.
"It's also to ensure that they can stay at work and be that first line of defence against Covid and protect the patients they care for."
A Scottish government spokesperson said: "We know NHS Scotland boards have seen increased and earlier demand this year for the vaccine from eligible groups, including health and social care staff. Boards are responsible for delivering the vaccine this year including managing their allocated vaccine supply in line with demand within their areas.
"The Scottish government has procured sufficient vaccine to vaccinate all those who have been invited to receive the seasonal flu vaccine, in line with uptake planning assumptions of 2.4 million people, and we are working closely with National Services Scotland to ensure that boards can access this stock as required."