The council would maintain a smaller presence in Chester through its existing offices at Nicholas House and Goldsmith House but it is unclear whether the headquarters function would remain here.
It also unknown what the move would mean in terms of the many council meetings, open to the public, that are held in HQ such as planning committees and cabinet meetings.
When CWaC was first set up in 2009 it was agreed to hold full council meetings at Wyvern House in Winsford but to make Chester the administrative headquarters.
The latest move could further undermine the status of Chester as the county town following the migration of the police and fire headquarters to Winsford several years ago.
Under the proposals, some staff from HQ would be relocated from and to Wyvern House and Civic Way in Ellesmere Port in a plan that would include improvements to both corporate buildings.
The vision includes an allocation based on one desk per 1.8 people, an offer of full-time home working for some staff and hot desk provision.
All council employees based in corporate buildings are being asked for their views on the ‘agile working proposals’ that would enable people to work from a variety of locations supported by technology.
A staff consultation runs until April 28. The views of unions and elected members will also be considered.
Chief executive Gerald Meehan said: “Advances in technology mean that people are working differently now and we need to look at better ways of working for our staff based in corporate buildings.
“It is also important to make accommodation cost savings if we are to achieve our aim of bridging the £57m financial shortfall in our funding by 2020.
“These proposals will place our services closer to our residents, enabling our local communities and economies to thrive.
“They will offer our staff a more flexible approach to working and help us to achieve our financial savings by operating out of our more affordable corporate buildings.
“This is a continuation of our journey to be more effective and efficient, delivering the best possible services for our residents.”
The proposals are expected to be finalised in the summer. If agreed, relocation of staff would begin in early autumn.