Sky Sports understands Graham is eager to return his native north east and has been heavily linked with a move to Sunderland.
Swans boss Michael Laudrup confirmed that the club had received an offer for the 27-year-old, but hinted that it did not meet their valuation.
Norwich are also reportedly interested in the striker, who has scored seven goals in 22 appearances this season.
Despite his desire to leave, Laudrup said that Graham could remain at the Liberty Stadium beyond the current transfer window.
He said: "If he comes and says he wants to leave and there's a very good offer we'll have to look into it.
"It could easily happen that we have exactly the same squad in February as we have now."
Graham has been included in Swansea's squad for Tuesday's Premier League trip to Sunderland.
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