In its statement, the council said Galliford Try was working through issues that needed to be addressed, including the legionella and a fault with the water system.
It said tests had confirmed the legionella had gone, but the water system would remain under Galliford Try’s control until all required works were fully completed, independently tested, and formally signed off.
A council spokesman told the BBC some adaptations were required to the water system to maintain cold and hot water temperatures within appropriate ranges.
The authority is working with Galliford Try and NHS bodies to agree a timetable for opening, but this depends on successful completion of tests as well as full handover documentation, he said.
Costs are being closely monitored, he added.
The council has said the health centre will be the first of its kind in Sussex, bringing together a range of NHS services under one roof and improving people’s access to care.