Andrew Bogut, not only looking for a larger than peripheral role but also the best chance at a return to the NBA Finals, turned down the Celtics last night and agreed to terms with the Cleveland Cavaliers.

The Cavs, who also signed point guard Deron Williams, will be a more significant challenge than ever for the Celtics, who are in second place in the Eastern Conference and on a slide with losses in three of their last four games.

And for the time being, the Celts don’t have what a source called a “Plan B” to improve their obvious weakness on the glass.

The source said free agent forward Terrence Jones is off the table, though management hasn’t ruled out adding Jared Sullinger, the former Celtic traded from Toronto to Phoenix and then bought out, “somewhere down the line, if it is needed.”

Sullinger, for all of his conditioning and health issues, did indeed make the Celtics a better rebounding team last season.

“But there’s nothing imminent, (the Celtics) will continue looking and see what happens,” said the source, who added that the Celts may be forced to improve their rebounding with existing personnel.

Avery Bradley, though a guard, is currently the team’s leading rebounder. His return after missing 22 games to a strained right Achilles will be, for the time being, the best short-term chance at a rebounding upgrade that they have.

Bogut, clearly, would have marked a significant upgrade on the glass. But according to a second league source, the Celtics debated the value the oft-injured Bogut could bring, both as a major part of the rotation and as a player capable of staying healthy. Bogut has been clear about his desire to play a significant role wherever he goes.

“At the deadline (the Celtics) had mixed reviews on whether he’d be a help or not,” he said.

The second source was skeptical of the Celts’ chances of signing Bogut, citing a disadvantage that turned out to play a big role — Cleveland’s overwhelming odds of returning to the Finals, thus giving Bogut a chance to get back at the Golden State team that traded him to Dallas last summer to make room for Kevin Durant.

The Celtics could have offered Bogut a pro-rated sum of $2.046 million under the so-called room mid-level exception, slightly more than the Cavaliers and Spurs, who can only offer minimum contracts.

The Celtics had the second most money to offer Bogut, who clearly valued a relevant role over money after getting left out of Mavericks coach Rick Carlisle’s rotation this year.

Bogut missed 16 games between December and January due to knee and hamstring issues, and only played once since January in a Feb. 15 game against Detroit when he had four points and eight rebounds.

Bogut’s injury history, and the possibility that the Celtics could lose him to an injury was a deterrent. There was also some debate over just how much of a role Bogut would play. He would immediately shore up two areas — rebounding and rim protection — where he would instantly become the best on the Celtics in those two roles.

The debate arose over whose minutes would be sacrificed to accommodate Bogut, starting with the inconsistent Amir Johnson. Coach Brad Stevens has come to depend on Kelly Olynyk’s scoring and role off the bench, and his on-court time would be unlikely to decrease with Bogut’s arrival. The issue would be where else to find those minutes, and what role players would be affected, a job further complicated by Bradley’s return.

The Celtics, despite their obvious needs for interior muscle, are also striving to use the kind of small ball lineups that Golden State perfected. They now may have no choice.