Martin's initial month-long deal with the Rams ended following their 2-1 home win over rivals Leicester on Saturday evening.
"We are trying to keep Chris. We have been trying in the past week to sort something out and, hopefully, we are not too far away," said Rams boss Nigel Clough following the victory over Leicester.
"I think he wants to stay and that is the main thing. He has done well for us. He is a calming influence. He gets hold of the ball, keeps it simple and retains possession."
Asked if Martin's loan could lead to a permanent deal for next season, Clough added: "That will purely depend on what Norwich's plans are for him."