The England centre knows the decision is out of his hands but he wants to repay the club for signing him when he was in the Leicester second team.
Quins lost to Sale on Saturday and missed a last-minute penalty. Had Jeremy Staunton converted it Northampton and not Quins would have gone down.
"It may not be down to me but as far as I can say if Quins require my services I will be with them next season," Greenwood told the Daily Mirror.
"They have been very good to me over the years. They showed an enormous amount of loyalty when rescuing me from the Leicester second team.
"Now it's my turn. I love London and I've loved just about every minute of my time at the club though this obviously is a terrible time."