Steve McClaren's club have been lining up a move for Shelvey during this month's transfer window to bolster their squad and have now firmed up their interest by making an approach for the England international.
Newcastle are willing to pay £10 million for the 23-year old, Press Association reported.
Shelvey has fallen out of favour at Swansea in recent weeks losing his place under first Garry Monk and now under new boss Alan Curtis.
The former Liverpool man returned for the FA Cup defeat to Oxford on Sunday before becoming embroiled in more controversey by appearing to be involved with an angry confrontation with a fan.
Newcastle boss McClaren is a big admirer of Shelvey and sees him as someone who can play a key role in helping his side in their battle to avoid relegation.