The winners of City's rearranged match at Oxford United next Tuesday (31 January 2017) will play away to either Luton Town or Yeovil Town in the Checkatrade Trophy semi-finals.
Thursday's last-four draw ensures Stuart McCall's Bantams or Oxford will face Sky Bet League Two opposition for the chance to play in the Wembley final in April.
Like City's original game with Oxford, Luton Town's quarter-final tie at home to Yeovil Town this week was also postponed due to a frozen pitch.
The Bantams will travel to Oxford next Tuesday (31 January) to try again in their last-eight encounter.
Luton and Yeovil have rearranged their own quarter-final tie for Tuesday 7 February.
The other semi-final tie will see Coventry City host Wycombe Wanderers.
Semi-final ties are scheduled for the week commencing 20 February.