The 18-year-old has signed a three-and-a-half year deal at the Liberty Stadium having completed a medical on Monday.
He becomes Swansea's second January signing following the arrival of striker David Ngog from Bolton.
The Welsh club are also waiting to find out if they have landed Tom Ince on loan after holding talks with the Blackpool winger on Monday.
King, a Scotland Under-19 international, told the club's official website: ""It feels brilliant to be here.
"I've left a passionate club and I'm arriving at another passionate club.
"It's always going to be hard to leave the club that you love, your home and your family as well, but coming to Swansea, where they're building something special here, it's a very good club to be at.
"The way Swansea play football is brilliant and will suit me down to a tee. I like to get the ball down and play, not force passes, and get around the pitch.
"For me, being a central midfielder, I admire the likes of Jonjo Shelvey and Leon Britton here, and I believe I can learn a lot from them.
"I'm excited and I'm looking forward to my time here."
King took part in his first training session with the Swans' Under-21 squad on Tuesday morning.
Swansea have also allowed young left-back Jernade Meade to join Luton on loan for the rest of the season.