Ronny Deila stressed his desire to keep Fraser Forster at Celtic following the 4-0 victory over KR Reykjavik.
Executive vice-chairman Ed Woodward insists Manchester United are ready and willing to break the British transfer record.
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."