The former West Brom captain completed a protracted move to Craven Cottage in August.
Greening told fulhamfc.com: "I'm really, really enjoying it and the lads have made me feel really welcome. I'd obviously like to stay as long as I can because it's a great Club and a great place to be.
"When you come to a new Club you want a run in the team as quickly as possible so you can get to know your team-mates. Danny [Murphy] obviously got injured unfortunately and I've just tried to come in and do the best that I can. I still think I can do a lot better and hopefully I can have a longer run in the Team. The longer I play and get used to my team-mates the better I will play.
"The ambition is to finish as high as possible in the Premier League and qualify for the next round of the Europa League. If we can finish in the same way we did last year it would be an excellent season but it's a difficult League. The most important thing is to cement your position in the League. I don't see why we can't finish like last season because we've got the players to do that. "