Adam Himmelsbach of the Boston Globe reported Bogut is scheduled for a bone scan Tuesday and is optimistic about signing a contract next week.
The 32-year-old Australian signed with the Cleveland Cavaliers in early March after getting waived by the Philadelphia 76ers following a trade from the Dallas Mavericks. He suffered the injury just a minute into his tenure with the Cavs.
Health setbacks were a major issue for Bogut during his time with the Milwaukee Bucks early in his career, but he showcased better durability with the Golden State Warriors. He played at least 67 games in each of his final three seasons with the Dubs before joining Dallas.
He can still be an effective interior presence, having averaged 10.0 points, 8.9 rebounds and 1.6 blocks across 671 career games.
In June, Bogut told Steve Aschburner of NBA.com he believes there's still a niche for old-school big men despite the trend toward post players who can stretch the floor offensively.
"If you look at a lot of the teams with stretch 5s, a lot of them have rebounding issues generally," he said. "I know it's not a max market for big guys, but you can still make a decent living from doing the little things. I like to think I do a lot of good things that go unnoticed as well. Teams I've played for, teams that have played against me know the value I have."
Himmelsbach didn't identify which teams were interested in Bogut, though he reported in August the Boston Celtics had been "in contact" with the center's representatives.