Knill is rated as one of the most promising managers in the lower divisions and took the Shakers to the play-off semi-finals in the recent campaign.
The former Rotherham boss was linked with a move to Blackpool before Ian Holloway took the job and told the Bury Times: "The board have made me an offer and I am mulling it over.
"I see this as work in progress. There are things that needed to be done better when I came here, and there are still things we need to work on.
"It takes time to improve a club and hopefully I will get that. I feel we have brought the club forward quite a lot in the time I have been here."
Knill is expected to announce the signing of Chester striker Ryan Lowe on Monday.