After his trip to Chicago, Anthony reportedly will head to Texas on Wednesday to meet with the Houston Rockets and Dallas Mavericks.
Anthony, who has an offseason home in Southern California, also expects to meet with the Los Angeles Lakers next week.
Despite that, the Knicks remain confident that they can re-sign the All-Star. ESPN quoted a source as saying team president Phil Jackson has "connected" with Anthony.
Jackson reportedly is planning to recruit his former Lakers center, Pau Gasol, to replace recently-traded Tyson Chandler and pair him with Anthony on the Knicks' front line. However, the Knicks can only offer Gasol less than $4 million for next season.
Gasol will also become a free agent Tuesday.
The Knicks sent Chandler and point guard Raymond Felton to the Dallas Mavericks last week. The Knicks received point guard Jose Calderon as part of the deal with the hope that Anthony would look forward to teaming with the Spaniard.
The Knicks can offer Anthony a maximum contract of five years and $129 million. The most any other team can give him is four years and $96 million.
Anthony is reportedly intrigued by what the Bulls have in talent, including point guard Derrick Rose, center Joakim Noah and coach Tom Thibodeau.
However, Anthony has said that he is reluctant to uproot his family, which includes his 7-year-old son, to another city.