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Last week the Celtics were prepared to enter training camp with 16 players under guaranteed contracts, one above the league maximum. It would have been the third consecutive season in which they started camp with a surplus.

But everything changed Tuesday night, when the Celtics acquired All-Star point guard Kyrie Irving from the Cavaliers in the blockbuster trade of the summer. In return, Boston parted with All-Star point guard Isaiah Thomas, forward Jae Crowder, rookie center Ante Zizic, and the 2018 first-round draft pick the Celtics had previously acquired from the Brooklyn Nets.

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Suddenly, the Celtics had an open roster spot. This shift was good news for players such as Shane Larkin, Abdel Nader, and Daniel Theis, who now must feel more confident that they will make the team. It also made Boston a possible destination for free agents looking for work.

A league source said Boston will not rush to fill the spot, and might even keep it open to maintain flexibility. The Celtics this week have been in contact with representatives of several free agents. According to another league source, that group includes center Andrew Bogut, who considered signing with Boston last February before ultimately agreeing to a deal with the Cavaliers. Bogut broke his leg in his first game with Cleveland, though, and has spent this summer rounding back into form.

The Celtics are over the salary cap, so they will be allowed to offer just a minimum-salary deal that is slotted based on years of experience. That figure maxes out at $2.3 million for a player who has been in the league for at least 10 seasons.

Celtics second-round draft pick Jabari Bird, who showed flashes of offensive potential when he shot 61 percent from the field in the Las Vegas summer league, is still unsigned. But a league source said it remains most likely that Bird will receive a two-way contract that would place him with the G-League’s Maine Red Claws and allow him to spend up to 45 days with the Celtics.

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But there are plenty of capable veterans on the market, and the Celtics could even sign two of them if they wish to enter camp with 16 players as originally planned.

Despite making some franchise-altering moves this summer, the Celtics did little to resolve their lingering rebounding issues. They ranked 27th in the NBA in rebounding percentage last season, and four of their five top rebounders are gone.

With five rookies on the roster and just four returning players, Boston could also look to an experienced veteran to serve as a mentor and locker room leader. Here are some players the Celtics could sign.

■  Andrew Bogut, center

It’s hard to believe that Bogut is just over a year removed from being the starting center on the Warriors’ 73-win team. Last season he struggled in Dallas before being traded to the 76ers in February. After being bought out by the 76ers, Bogut signed with the Cavaliers rather than the Celtics.

Still, his discussions with president of basketball operations Danny Ainge and coach Brad Stevens resonated, and a source close to Bogut said he is very interested in joining Boston and is expected to be fully healthy at the start of training camp.

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Bogut is a strong defender and, if healthy, he would immediately become the Celtics’ best rebounder. But Boston would want to be sure that the 32-year-old has sufficiently recovered from the broken leg that sidelined him for several months.

■  Tony Allen, guard

The former Celtic turns 36 in January, but he played a career-high 27 minutes per game for the Grizzlies last year. Allen remains a pesky defender, and with 23-year-old Marcus Smart suddenly the Celtics’ most tenured player, the return of a member of Boston’s 2007-08 title team would bring a jolt of nostalgia, perspective, and leadership. Young wings like Jaylen Brown and Jayson Tatum could soak up his defensive wizardry.

If fans could pick, Allen would certainly be the top choice.

■  Gerald Green, guard/forward

The Celtics selected Green in the first round of the 2005 draft, and he spent two seasons with them before being traded to the Timberwolves in the Kevin Garnett deal. Green returned to Boston on a one-year minimum-salary deal last season, and he was an essential locker room presence. He kept the mood light with his personality and served as a kind of assistant coach.

Green showed he still has value on the court, too. After the Celtics dropped their first two opening-round playoff games to the Bulls, Green entered the starting lineup and Boston won four in a row, including Game 4, when Green tallied 18 points and 7 rebounds.

■  Other free agents to watch: forward Boris Diaw, forward Dante Cunningham, forward Thomas Robinson, forward Kris Humphries, center Tyler Zeller, forward Matt Barnes.

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Adam Himmelsbach can be reached at adam.himmelsbach@globe.com. Follow him on Twitter @adamhimmelsbach