The 34-year-old Carter, who recorded 10 sacks in 14 games for the Patriots in 2011, had a workout with New England last month.
"We have a lot of respect for Andre. He's a hard-working guy who has given us great leadership in the past. He's a good teammate, everyone loves Andre," coach Bill Belichick said Wednesday. "In the end, it still comes down to performance on the field and all that. We'll see."
Carter had his 2011 season cut short due to a quad injury. There was mutual interest between him and the team for a reunion last offseason, but Carter eventually chose to play for the Oakland Raiders. He had 2.5 sacks in 12 games.
He was released by the Raiders before this season began.
The Patriots waived defensive tackle Andre Neblett on Monday, leaving an open spot on their roster.
With captains Vince Wilfork and Jerod Mayo landing on season-ending injured reserve, the Patriots not only lost two of their best players, but also a steadying veteran presence. Carter offered a similar-type veteran presence in his one season with the Patriots.
ESPN's Josina Anderson and ESPNBoston.com's Field Yates and Mike Reiss contributed to this report.