Boss John Askey is a huge admirer of central midfielder Norburn, 25, who was left out of Micky Mellon's Tranmere squad for their League Two opener against Stevenage on Saturday.
But Sunderland will rival Town for the signing of Norburn. It is believed Rovers chief Mellon wants a sizeable offer for one of his prize assets.
Norburn has previously shone under Askey while at Macclesfield and played a key role in Tranmere's promotion from the National League last season.
Town have been pursuing the midfielder all summer as one of the boss's top targets and the race looks set to continue with Thursday's deadline looming.
Funds could be available for Askey with midfielder Jon Nolan and defender Toto Nsiala on the verge of sealing moves to Ipswich.
Shrewsbury are also keen on a loan deal for Rangers midfielder Greg Docherty. It is believed The Black Cats, managed by Scot Jack Ross, are also interested in taking Docherty.