Reports from Scotland have suggested the Hoops are ready to join Ipswich, Reading and Championship play-off finalists Cardiff in monitoring his situation as he looks for a move away from the Stadium of Light, and the Wearside club is aware of the mounting interest.
The 27-year-old Republic of Ireland international has two years remaining on his contract, but has failed to win a regular place in the starting line-up at Sunderland.
Darren Bent and Kenwyne Jones were the men in possession at the end of last season, with summer signing Fraizer Campbell also playing his part, and while loan signing Benjani Mwaruwari will not be returning, youngster Martyn Waghorn might following a successful campaign on loan to Leicester.
Murphy impressed during a loan spell at Ipswich during the last campaign, where he scored six times in 18 appearances for Roy Keane's side.
However, he is one of a series of players, along with the likes of Phil Bardsley, Nyron Nosworthy, Marton Fulop, Anton Ferdinand, Teemu Tainio, George McCartney and David Healy, who could leave the club this summer as manager Steve Bruce shuffles his pack.
The former Waterford frontman joined Sunderland in a £100,000 move in June 2005 and his contract has been improved significantly since, meaning he could have to take a wage cut to secure a deal with another club.