Allardyce has signed a two-year deal with the shakers on a scholarship, with plans to force his way into the first team squad.
Speaking to Bury's official website, Allardyce noted how the club could be primed for him to accelerate his career and development as a footballer.
"With there being no Under 23's team, it means that the opportunity is there to get noticed earlier, as some Under 23's enter a team and never get seen again," the youngster said.
"I want to keep playing in my position and constantly improve.
"Hopefully I will impress early, and get constant starts for the youth team with an opportunity to train with the first-team."