Backup forward Jeff Green, who missed all last season with an aortic aneurysm that required surgery, and free agent guard Jason Terry still haven't signed but were at a news conference with Bass, Wilcox and Celtics president Danny Ainge.
The deals for Green and Terry could be signed next week, but the delays are the result of the team's manoeuvring under the new collective bargaining agreement.
"We hope to resolve those sometime in the near future," Ainge said. "Things are moving forward."
Garnett, the Celtics' emotional leader, is the most important player returning, especially after Allen's decision to leave as a free agent broke up the Big Three of him, Garnett and Paul Pierce that won one championship in five seasons together. Last season, the Celtics reached the Eastern Conference finals where they lost to eventual champion Miami in seven games.
"I don't feel like we lost Ray," Ainge said. "I feel like Ray left us to go to Miami."
His primary target, though, was Garnett, who was offered a three-year contract while Allen reportedly was given a two-year proposal.
"That was a priority for us this summer, to get KG back," Ainge said. "That was a great relief to get him back. He's such a big part of who we are and going forward we think he has a lot of basketball left in him."
Garnett is the most important part of the team's future after having an outstanding season, especially down the stretch and in the playoffs. He averaged 15.8 points, 8.2 rebounds and 2.9 assists.
Green said he always intended to stay with the Celtics.
"With the stuff I went through the past couple of months, this organization was there for me," he said. "They did everything possible to make sure I was comfortable with my procedure and the road I was headed down. I owe it all to them. It was a no-brainer."
Wilcox had surgery for an enlarged aorta after playing 28 games, averaging 5.4 points and 4.4 rebounds.
"It means a lot just to come back and be able to play," he said.
Bass averaged 12.5 points and 6.2 rebounds while starting 39 of his 59 games after being obtained from Orlando.
"I knew where I wanted to be," he said. "That was back in a green jersey."
Terry spent the last eight seasons with Dallas, averaging 15.1 points in 2011-12. In 13 seasons, he is averaging 16.1 points and figures to replace Allen as the team's primary three-point shooter.
"If they need me to wash the uniforms or sweep the floors, whatever role (coach) Doc (Rivers) wants me to play I'm here to do it at 100 per cent and give my all," Terry said.