Demolition of the Oceana building on Kingsway is taking place but a digital village is due to go there instead as part of the Swansea Bay City Deal.
The authority has now turned to the car park on nearby Oxford Street which is likely to be the new location.
It needs a new base as the civic centre on the seafront will make way for the Â£500m city centre regeneration scheme.
"Our civic centre plans mean council staff will have to be relocated to a new building in the city centre, with alternative car parking for staff and Grand Theatre visitors also being looked into," said council leader Rob Stewart.
"No final decision has been made, but the Oxford Street car park site has been identified as an option for a replacement building for council staff as we look to further boost city centre footfall and encourage more spending in our city centre businesses.
"Relocating our staff to the city centre could boost the city centre economy by Â£5m to Â£6m a year."
The current civic centre site is likely to be replaced with public spaces, and tourist attractions including an aquarium and a digital science centre.
The council bought the Oceana building - which was once at the heart of the city's nightlife - in 2015 and the site was looked at as a new home as part of creating a business district.
But now it has revealed plans to go elsewhere.
Opposition leader Chris Holley is against the move.
"My own personal opinion is the civic centre should stay where it is," he said.
"There's no reason why they shouldn't develop one of the car parks next to it and we can also keep one of the best-used central libraries in the country."
Plans to turn the site of Swansea's biggest nightclub into a new home for the city's council have been scrapped.