A £350,000 fee is being talked about for the 25-year-old, currently playing a key role in Sheffield United's promotion challenge in League One but possibly available in January.
Quinn is already a veteran of over 200 league games having moved to Bramall Lane from his native Ireland.
Interest from a number of clubs in the player was mooted at the end of the summer after the Blades' relegation from the Championship with a bid, reportedly from Blackpool, turned down in the final days of the window.
Forest take on Leicester City in the FA Cup this weekend - check out the head-to-head record of the East Midlands' rivals by clicking the link below.