Manchester City and Newcastle United are both long-term admirers of Allardyce and could make a move for him at the end of this season.
Allardyce said: “This has been one of the hardest decisions I’ve ever made but it’s time for a new exciting era for Bolton under different leadership.”
Chairman Phil Gartside added: “His reasons are private, we respect that.”
Allardyce said in a statement on the club website: “It was mutually decided that I step down now rather than at the end of the season.
“That will give my successor the ideal opportunity to experience the preparation and build-up of match days in readiness for next season.
“After guiding the team into a position where the club is on the verge of qualifying for European football for the second time in three years, I feel it is the right moment to step down.
“I feel the club is in a great position to continue its upward trajectory, with a talented squad supported by an excellent Academy programme under the guidance of a highly-skilled management team.
“The foundations for the next phase are well and truly in place. It is with this knowledge that I feel confident that I am moving on with my life, content with a legacy that I have tried to create for this special club.
“I wish the owner, chairman, staff, players and all the fans every success in the future.”