Albion have been drawn at home to either Brentford or Wolverhampton Wanderers in the fourth round of the Emirates FA Cup.
The Baggies eased to a 4-1 triumph at home to Vanarama National League outfit Aldershot Town in round three on Sunday, while the Bees and Wolves played out a 1-1 draw in London on Friday night.
Their third-round replay is due to take place at Molineux on Tuesday, January 16.
Albion last faced local rivals Wolves in the FA Cup in January 2007, winning 3-0 away from home thanks to efforts from Diomansy Kamara, Kevin Phillips and Zoltan Gera.
The Baggies have only ever met Brentford once in the competition, winning 2-0 at The Hawthorns in January 1959.