Thr second division, currently fighting for automatic promotion, side are keen to finalise a deal for Gillespie before other interested parties swoop for the exciting forward.
Bristol City chairman Steve Lansdown said: "We will be getting together with Steven and his agent to talk personal terms and that is likely to happen next week.
"We understand other clubs are interested. But we have done things right and I think he sees an opportunity to challenge quickly for a first team place.
"I'm hopeful we can reach agreement. If he does sign, supporters should understand that he is one for the future and not expect him to be banging in goals at senior level straight away."