The former West Ham trainee had been linked with a move away from the Swans in the summer.
But Britton told BBC Wales Sport: "The talks (over a new contract) have been going well. The ball's back in the club's court.
"We're just waiting for a phone call from them to let us know a few things and then we'll go from there. Hopefully things will be tied up."
The 26-year-old added: "I'm very keen to stay. I'm happy here.
"I wouldn't have talked to the club about a new contract if I wasn't happy - I don't think that's the right thing to do."