Like many of City's players the former Cardiff shot-stopper is out of contract in the summer and faces an uncertain future.
But Heaton, one of the club's best performers this season, has admitted he would consider staying if the Robins offered him a deal.
"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."
City were relegated from the npower Championship following the home defeat to Birmingham and are already thinking about life in the third tier.