The Black Cats made an offer on Thursday, believed to be above the ?4million mark, and the Canaries are understood to have followed suit.
Graham has scored four goals in his last five appearances for Swansea, but has been frustrated at his lack of game time thanks to the goalscoring form of Michu.
The 27-year-old, who is thought to prefer a return to his native north-east, started his career at Middlesbrough, before his impressive form for Carlisle and Watford yielded a £3.5million move to Swansea in the summer of 2011, with the fee making Graham the club's then-record signing.
He went on to score 14 goals in 39 appearances in his debut season, and has struck seven times in 21 outings this term.
When asked about possible January signings on Thursday, Canaries boss Chris Hughton said: "Numbers-wise we're fine as regards the squad. The injury situation hit us a bit up front having Grant Holt and Steve Morison out for a period of time.
"Can we add to it? If we are able to add some quality to what we've got we'd gladly do that. We are ambitious here. We do want to develop but it has to be in the framework of the club."
On a link with a move for Celtic forward Gary Hooper, he added: "My responsibility is to report on anything that is fact or almost fact and we are not at that stage at the moment."
With Graham potentially leaving the Liberty Stadium, Swansea could be set to firm up their interest in Stoke forward Kenwyne Jones, with reports suggesting they are primed to return with an improved offer having had a £3million bid turned down.