The 29-year-old former West Brom man, who missed the whole of last season through injury, has been a regular for the Robins and scored his fifth goal of the season in Saturday's 1-1 draw with Port Vale.
"I have enjoyed every minute of my time here and I would certainly have no qualms about staying longer," Carter told the Gloucestershire Echo.
"I will be speaking to the gaffer over the next few weeks definitely, having a sit down and a chat and we'll see where we go from there.
"The club have looked after me and I just wanted to get back playing so I've loved it. It's a great little club that have given me that opportunity and the gaffer and Neil (Howarth) have been fantastic."