Swansea Council leader Rob Stewart said any approved plan would accommodate the relocation of Swansea Cricket Club.
They have played at the venue since 1875 but Glamorgan have not been there since 2019.
“St Helen’s needs to be upgraded to offer a modern sports experience, and the Ospreys need a new home and a plan for long-term sustainability,” said Stewart.
“We’re prepared to work on the current Ospreys proposal for St Helen’s to become a modern rugby stadium, helping the region to make their home there and remain in Swansea.
“We’re doing all we can to support the Ospreys to stay in Swansea, while also working with all of our sporting stakeholders to provide top-class facilities for them.
“We’re already discussing key matters in a positive manner with current St Helen’s residents Swansea CC and Swansea RFC.
“Ongoing discussions to enable these potential changes include implications, timescales and funding to develop improved cricket facilities nearby.”
Councillor Neelo Farr, Cabinet Member for Regeneration, Economic Development and Housing for Bridgend County Borough Council says an Ospreys move to the area would “provide a massive economic boost”.
“We are fully supportive of the Ospreys’ plans to potentially relocate to Bridgend County Borough and we are looking forward to further talks as we continue to discuss the many mutual benefits of any such move,” said Farr.
“The Dunraven Brewery Field would offer the Ospreys a unique opportunity to base themselves in the heart of a town centre, with Bridgend being perfectly placed on the M4 corridor and having lots of public transport options with nearby train and bus stations.
“The Ospreys already have lots of strong community links throughout the county borough and it’s fantastic many of their current players are from Bridgend and started their rugby journeys by playing for our local teams.”