On Tuesday, Adam Himmelsbach of the Boston Globe reported Bogut is expected to return to the United States from his native Australia over the weekend to sign a contract next week.
Matt Logue of the Daily Telegraph added the San Antonio Spurs, Milwaukee Bucks and Utah Jazz are also believed to have reached out to the center's representatives.
The 32-year-old University of Utah product averaged 3.0 points and 8.4 rebounds across 26 games for the Dallas Mavericks last season before getting traded to the Philadelphia 76ers in February. The Sixers waived him less than a week later.
Bogut signed with the Cleveland Cavaliers in early March to provide depth during the team's repeat bid, but he suffered the season-ending leg injury a minute into his Cavs debut.
In June, the veteran center told Steve Aschburner of NBA.com he understood the league has moved more toward post players who can stretch the floor offensively, but he felt his game could still bring value.
"If you look at a lot of the teams with stretch fives, a lot of them have rebounding issues generally," Bogut 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."
The Aussie has averaged 10.0 points, 8.9 rebounds and 1.6 blocks in 671 career appearances.
Boston could use some additional frontcourt options behind the projected starting tandem of Al Horford and Marcus Morris. Bogut could be a defensive stopper for a squad with a lot of offensive talent.
The same could be said for the Spurs, who don't have much in terms of experience to provide reliable minutes when LaMarcus Aldridge and Pau Gasol are on the bench.