Rose was part of the side which won promotion from League Two at Wembley in 2008 and will compete with Adam Drury for a starting role at Carrow Road.
He joins former Hatter Oli Johnson in moving from Stockport to Norwich this month.
Meanwhile, City full-back Rhoys Wiggins is joining League Two side Bournemouth on loan for 28 days after failing to break into Paul Lambert's plans.
Source: Team Talk
