The Ashton Gate club had proposed staggered kick-off times considering neither fixture is considered low risk, let alone high risk.
Both teams have Johnstone's Paint Trophy ties the previous Wednesday - City are at Nottingham Forest while Rovers host Peterborough.
They also have league fixtures the following Tuesday - City host Carlisle while Rovers travel to Lincoln.
So neither club wants to play on Friday or Sunday.
Rovers believe they have the choice as City would have had priority if both of the clubs' first-round ties had gone to a replay in Bristol.
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