The 62-year-old managed the Trotters between 1999 and 2007, turning them into an established Premier League side and guiding them to a League Cup final and European qualification.
Allardyce spoke highly of Bolton, describing them as "the club that gave me everything", but believes that they could have gone further with more ambition from the board, citing Premier League champions and Champions League quarter-finalists Leicester as a case study.
The Crystal Palace boss told the Evening Standard: "My greatest club I managed was Bolton because it was the club that gave me everything. But when it told me it had no ambition, I left.
"I did not want to leave but they did not want me to get in the Champions League or win any cup competitions. They just wanted to stay in the Premier League.
"That is no good for me. You do not do this job to achieve mediocrity. You do it for success and getting as high as you can. Look at Leicester. In my time that could have been Bolton."
The Macron Stadium outfit slipped into League One at the end of last season but currently sit second in the table, seven points clear of third-placed Fleetwood Town with four games to go.