Maynard has been in negotiations over a new deal since the summer, with his contract expiring at the end of the season.
City reportedly turned down an offer of £6million from fellow Championship club Leicester City in June.
Chairman Colin Sexstone told BBC Radio Bristol: "At the moment he doesn't want to discuss contracts and because we are getting close to the window, we change our tack slightly.
"Whereas before we were doing everything we could not to encourage offers for Nicky, clearly now time is pushing us and we will now listen to offers."
Bristol City paid £2.25 for Maynard when they signed him from Crewe in 2008.
Nottingham Forest, Southampton and West Ham were also interested in the striker in the summer and may come back in for him in January.