O'Neal had said earlier Tuesday that he hopes to make a decision about his future soon, and both NBA.com and Comcast SportsNet New England indicated that the Celtics were "close" to signing the four-time NBA champion. Later Tuesday, The Associated Press cited a person with knowledge of the negotiations as saying the Celtics were making progress on a deal for O'Neal.
The Atlanta Hawks had been the front-runner to land O'Neal last month and are not completely out of the picture, sources told Broussard, but they have fallen behind the Celtics.
O'Neal was not willing to reveal which team he was leaning toward.
"For me, it's been a real thinking process," O'Neal told The Associated Press on Tuesday from Orlando, Fla., where he was promoting his television series, "Shaq Vs.," with teen pop star Justin Bieber. "I came into the league very graciously and want to go out very graciously. My main thought was I would like to play for a winning franchise, somebody that's used to winning, somebody that keeps winning. Hopefully, I'll make my decision here in the next one or two days."
The Celtics can offer only the veteran minimum salary ($1.35 million for a veteran of 10-plus NBA seasons), but Cleveland could facilitate a sign-and-trade with Boston if the 38-year-old center requires more money. O'Neal has made nearly $291 million in basketball contracts alone, including $20 million with the Cavaliers last season.
O'Neal has found a quiet market for his services this offseason, which led some to speculate he might find a better payday overseas. But O'Neal indicated Tuesday that he plans to stay on this side of the Atlantic.
"I know I will play in the NBA next season," said O'Neal. "International? No. The good thing about me and my career is I came in, did it my way, did more than expected. For every athlete or every great person, there's a time when it has to end. When and if it ever ends it will be very graciously.
"We're going to have a big party. We're going to have a ceremony and the next thing I'll be waiting for is the entrance into the place where your name will never be forgotten."
The Celtics will start the season without center Kendrick Perkins, who underwent knee surgery last month and is expected to be sidelined into February. Boston utilized the mid-level exception to sign free agent Jermaine O'Neal last month, but Celtics president of basketball operations Danny Ainge often noted the team remained in the market for another impact big man.
No player on the free-agent market probably offers more potential impact than Shaq, who has averaged 24.1 points and 11 rebounds per game over 18 NBA seasons.
O'Neal appeared on ABC's "Jimmy Kimmel Live" on Monday and said he'd like to play two more seasons in the NBA and that the Celtics would potentially be a good fit for him.
O'Neal stressed to Kimmel that he's on a "730-day program," suggesting he wants to play two more years until he reaches 40. When Kimmel suggested he couldn't picture O'Neal in a Boston uniform, O'Neal responded, "The Celtics would be a good program."
Pressed if he would feel weird putting on a Celtics jersey after spending time with the Lakers, O'Neal simply shook his head and said, "No."
The Celtics have aimed to sign free agents to short-term deals with an eye toward a massive roster turnover following the 2011-12 season. Even by signing O'Neal to a two-year deal, the team would not be jeopardizing its future plan.
Information from ESPNBoston.com's Chris Forsberg and The Associated Press was used in this report.