The Chicago Bulls player most likely to have a breakout year this season is Taj Gibson. Gibson is entering his fourth season in the league, and the final year of his rookie contract. NBA.com's Steve Aschburner wrote that Gibson is the Bulls' most intriguing player this year, and SB Nation's Blog A Bull agrees.
In the wake of Omer Asik's departure and Carlos Boozer's descent (that last bit is my editorializing), it's possible this year for Gibson to break out.
Gibson averaged 7.7 points, 5.2 rebounds, and 1.3 blocks per game last season (numbers that extrapolate to 13.6 points, 9.3 rebounds, and 2.3 blocks per 36 minutes) and should see more minutes and become more of a featured player in the Bulls offense this season.
If the Bulls don't sign Gibson to a deal by the end of October he will become a restricted free agent next summer and command a large salary similar to what Asik got this summer. The Bulls already have $58.2 million committed to Derrick Rose, Carlos Boozer, Luol Deng, and Joakim Noah for the 2013-14 season; so finding room for Gibson might be difficult, but Blog a Bull hopes Chicago can find a way.