It is understood the 21-year-old is due in South Wales at lunchtime to discuss personal terms with Swans officials.
Premier League newcomers Crystal Palace appear to have missed out on the player as their bid was £1million less.
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."
Swansea yesterday signed midfielder Alejandro Pozuelo from Real Betis, becoming the third Spaniard to sign in the past week after former Betis team-mate Jose Canas and Espanyol's Jordi Amat.
The club yesterday also brought in Gregor Zabret from Slovenian side NK Domzale and Cypriot Alex Gogic from Olympiakos on a free transfer, with both teenagers expected to go into the reserves.