A protest was held outside Edinburgh City Chambers yesterday, ahead of a meeting to discuss the handover.
Lothian Buses is one of the last remaining municipal bus companies in the UK.
Workers fear that if TfE take control of the firm it could pave the way for its sale to a private company.
They are also concerned that profits from the company could be used to fund the £145 million tram extension.
Unite regional officer Tony Trench said: “Lothian Buses is a jewel in the crown. It is a profitable business that provides a good service in Edinburgh and beyond. We can’t afford to have it threatened.
“Our members are afraid that Lothian Buses would be bled dry here under Transport for Edinburgh - either by privatisation, or because it’s being used as a golden goose to fill a hole in other budgets.”