Ipswich boss Mick McCarthy has offered a short-term contract to former West Ham midfielder Jack Collison.
Chairman Fawaz Al Hasawi has vowed to strengthen Nottingham Forest year on year, with loan arrivals possible.
The 33-year-old moved to the npower Championship strugglers from parent club Nottingham Forest last week on a deal keeping him in South Yorkshire until January 2.
But Greening, a former Champions League winner with Manchester United, is keen to stay and help the Reds avoid relegation.
"I want to play as many games as I can but I've got to do well to stay in the team," he told the Sheffield Star.
"If the manager is happy and wants to keep me until the end of the season and Forest are happy with that, I'd be delighted. I want to get some games under my belt. I'm going to knuckle down and give it my best shot."