Although they have featured recently in the reserves, the two defenders have yet to play for Stuart McCall's first team at the start of the 2016/2017 campaign due to injury.
McArdle spent the summer recuperating from shoulder and groin injuries while Clarke picked up a troublesome groin problem on the eve of the new season.
Fellow defender Matthew Kilgallon is set to join the pair in City's back four for the visit of the Shakers; the 32 year-old expected to begin the game at left back.
Daniel Devine is scheduled to replace Josh Cullen in central midfield following the West Ham loanee's call up into the Republic of Ireland U21s squad.
Attackers Haris Vuckic and Filipe Morais are likely to be named in the side as well. The game will come too soon for injured forwards Marc McNulty and James Hanson, however.
City assistant Kenny Black, who will lead the side on Tuesday night, said: "The lads who didn't feature at Chesterfield on Saturday will certainly come into the equation, as will some of the younger lads.
"The likes of Filipe Morais, Matt Kilgallon, Rory McArdle, Nathan Clarke and Daniel Devine, they need game-time so hopefully we can give them a run-out.
"It is slightly frustrating for them because of the form of the other lads currently in the side - they have been very consistent.
"The players waiting to get into the side have to be ready at the drop of a hat though, that is why these sort of games like Tuesday night are vital.
"It gives them a chance to show their form and to get some minutes under their belt."
Following City's win over Stoke in their opening group game at the end of August, victory for the Bantams against Bury would go a long way to securing their progression into the knock-out stages of the competition.
Bury also won their first group match though - 4-1 over Morecambe - and, like City, they have enjoyed a promising start to their Sky Bet League One season.
Black added: "It will be a challenging encounter for us.
"Bury have a good squad and they have been on a good run of form recently.
"With the top two qualifying from each group, a victory on Tuesday would put us in a very strong position. That is certainly our aim."