The Canaries have offered £5m for his signature, matching the figure reportedly tabled by Martin O’Neill.
Swans manager Michael Laudrup is believed to have deemed the 27-year-old surplus to requirements, and may be willing to do a deal in order to raise funds for other signings.
This has alerted Norwich, who are keen to strengthen their attacking options after a solid start to the Premier League campaign.
Newcastle, QPR and Reading are also believed to be interested, and could spark a bidding war.
Swansea are thought to have targeted Stoke’s Kenwyne Jones as a replacement, prompting talk that Graham may have to wait for a move.
The Potters will reportedly allow the Trinidad and Tobago international to speak to the South Wales club, although a deal is believed to be some way off.
Graham enjoyed a solid 2011/12 season, scoring regularly under former manager Brendan Rodgers, following his £3.5m arrival from Watford.
However, Laudrup appears to prefer Michu as his centre-forward, with the Spaniard having made an excellent start to life in the Premier League, netting 13 goals.
Graham has returned to the first-team recently, weighing in with a number of important strikes, including the club’s second at Stamford Bridge in the Capital One Cup.