The versatile left-footer, who joined from Nottingham Forest in 2003, was coming to the end of his contract at City and has been allowed to join George Burley's men for the remainder of the season.
Brennan never cemented his place in the side at Carrow Road, and becomes the second departure in a week after Dean Ashton's sale to West Ham. The club are hoping to announce some new faces before the end of the transfer window, with Colchester's Greg Halford hotly tipped to join.
Halford has been linked with a £1million move again in this morning's papers, although City are said to have been unsuccessful with a £2million bid for Cardiff's Cameron Jerome.