The on-loan Chesterfield defender joined the Pirates on an initial one-month deal at the turn of the year and has impressed in his five appearances for Rovers to date.
Manager Mark McGhee has already expressed an interest in keeping Downes at the Memorial Stadium, and the Australian himself would be delighted if that eventuality came to pass.
He told the Bristol Evening Post: "It's something we are going to sort out this week. At Chesterfield I was in the side for two or three games but then out again for the next two or three. It was very frustrating for me, but since I've come here I've played every game and I'm really enjoying it.
"It's a great club, everyone has been very supportive and they are a great set of lads too. If I can prolong my stay here then it's definitely something I'd like to do - whether that would be for another month, until the end of the season or permanently I'm not sure yet. We've not spoken about that.
"But staying here is definitely something I want to do."