The 24-year-old made 81 appearances for the Colls since joining the club in 2020.
Prior to this, the former Rochdale youngster enjoyed a spell at Chorley, before securing his first senior breakthrough in the Northern Premier League with Prescot Cables in 2018.
Lenehan signed for FC United of Manchester in May 2019 and made 38 appearances.
On his move to the Shakers, Lenehan said: "You only have to walk into the stadium to see the facilities. I spoke to Andy [Welsh] about the project and it’s something to be excited about and I’m looking forward to being a part of it.
"I’m looking forward to getting straight into pre-season and getting as fit as possible, and then it’s all about getting onto the pitch and hitting the ground running when the season starts."
Photo credit: Bury AFC
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