Sky Sports sources suggest the Canaries firmed up their interest in 27-year-old Graham on Thursday.
Sunderland are also believed to be in the running for the frontman, who has scored four goals in his last five games, having signed for the Swans in a £3.5million deal from Watford in 2011.
Newcastle, QPR and Reading could yet come into the reckoning, with Graham struggling to displace the prolific Michu in Swansea's starting line-up despite his recent positive run of form.
Graham may have to wait until a replacement is found for him before he can leave the Liberty Stadium, although Swans manager Michael Laudrup is understood to be lining up a move for Stoke's Kenwyne Jones.
The Potters striker is set to be given permission to talk with the south Wales club despite being a first-team regular at the Britannia Stadium.
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