The Swans have impressed in the Europa League this campaign, and while Williams admits that the club are trying to savour every moment of their run, he is keen to taste Champions League football before the end of his career, whether that be with Swansea or another team.
The 29-year-old told The Times: "I am completely happy at Swansea but I would like to listen to the Champions League music once in my life.
"If we can go to the next step, that would be the dream come true. We're probably seen as a little bit more established, the big teams probably see us as little old Swansea.
"Most of us have not played in Europe before and we are prolonging it as long as we can."
Swansea are currently in ninth place in the Premier League table.