City now have a new manager, former Yeovil chief Gary Johnson, and were expected to be amongst the front-runners in League One this season.
However, it has not materialised that way and the Robins' currently lie 19th in the table.
Notts will hope to take inspiration from last year's competition, where they managed to overcome Swindon Town, albeit with the aid of a replay. This led to a money-spinning third round clash at home to Premiership Middlesborough.
Gudjon Thordarson's side will make the trip to Ashton Gate - not a happy hunting ground - in two weeks' time.