The Barclays Premier League club currently share the council-owned stadium with rugby union side Ospreys through a 50-year lease.
Swansea were a third-tier side when the stadium opened in 2005 but they have since established themselves in the top flight and are keen to increase the capacity of the 20,800 venue.
"Positive, exploratory discussions with the Swans have now started about selling the Liberty Stadium to them," Swansea Council leader Rob Stewart told the South Wales Evening Post.
"This would give the Swans complete control of the stadium and further support their expansion plans in future, but it's important we protect the needs of the Ospreys too."
Stewart said Swansea Council had to make savings of £81 million over the next three years and the sale of the Liberty Stadium is expected to raise funds somewhere in the region of £20 to £25 million.
Swansea chairman Huw Jenkins, writing in the match programme for Saturday's 1-1 league draw with Sunderland, said the next few months "could change the way we operate forever."
"There are a lot of key decisions to make regarding the way this football club moves forward off the pitch," Jenkins said.
"Talks with Swansea Council about the ownership and running of the Liberty Stadium is top of the list, plus making sure the funding is right for us and in place to build the East Stand extension as soon as possible."
Swansea have held talks about potential external funding with American businessmen John Jay Moores and Charles Noell possibly taking a significant stake in the club for their investment.
But the Swansea City Supporters Trust, who own more than 21 per cent of the club's shares, are opposed to any change in the club's ownership.
"Things can never stand still, but risks have also got to be thought through to protect the club going forward," Jenkins said.
"Any capital investment off the field must never have a negative impact on it."
Source : PA
Source: PA