Sky Sports sources on Friday revealed last season's Capital One Cup winners had agreed terms with NK Domzale for the 17-year-old.
And Zabret has now revealed he chose a move to Swansea over their Premier League rivals West Bromwich Albion, as well as Portuguese giants Sporting Lisbon.
"It is true. I am coming to Swansea City," the teenager told Sportal.
"I got some offers this summer, even West Brom and Sporting Lisbon wanted me. But I chose Swansea City some time ago.
"That was a decision that was taken after talking with my family. I believe that my decision is the right one.
"I know I will have to work hard to prove myself but I am ready for that and I believe that I will succeed in Premier League."
Zabret will inevitably be one for the future for Swansea given they already have reliable goalkeepers in Michel Vorm and Gerhard Tremmel.