The former Manchester United stopper, who has been one of the club's star performers this season, is out of contract at the end of the season.
It had been reported that City's relegation to League One would see Heaton depart, but the 27-year-old has suggested that he could remain.
"Obviously, I'm soon going to be out of contract and everything is up in the air at the minute, but I would certainly consider staying if the club wanted me," he told the Bristol Evening Post.
"When things have settled down and the season is done and out of the way, we will sit down, have a conversation and see where we go from there.
"I have my own thoughts, but I would need to know how the club thinks about things. I will certainly sit down and have a good, long think about it and go from there. But the club has to make me an offer first."
Heaton has also played for Cardiff City.