The 21-year-old has agreed personal terms with Swans officials and has signed for four years.
Swansea chairman Huw Jenkins has spoken of the need to increase their quota of homegrown players and Shelvey, an established England Under-21 international who made his senior debut against San Marino last year, fits the bill.
"We have got to make sure that the make-up of our squad is right," Swansea chairman Huw Jenkins told the South Wales Echo.
"And obviously part of that is making sure that we have so many British players in there.
"We have got to do that - there is no choice because they are the rules."