Martin's initial month-long deal expired following Derby's 2-1 home win over rivals Leicester on Saturday evening, in which the Canaries striker scored his first goal for the Rams.
Clough hopes to have the 24-year-old back on loan when Derby resume Championship action after the international break at home to Bristol City on March 29.
"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 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."
Martin is out of contract at Carrow Road in the summer, although the Premier League club have a one-year option on the player.