Relegated from the English Premiership last season, Bristol currently top the Championship standings after seven wins from seven games and are widly tipped to gain promotion.
They are coached by former Crusaders player and New Zealand international Pat Lam, whose son Jack is the club captain, although he chose to represent Samoa at international level despite being born in New Zealand.
Bristol also have lock Joe Latta and No.8 Steve Luatua, an All Black international, as well as Tonga international Siale Piutau, who was born in New Zealand and is the brother of All Black Charles Piutau, who will join Bristol next season.
But last month, Carter suggested his future lay back home.
"I'm kind of going through those thoughts at the moment, whether I stay, whether I look at other opportunities," he told New Zealand's Radio Sport.
"New Zealand is where the family is and family is a big part of my life."