Martin's one-month deal is set to expire after Saturday evening's televised east midlands derby against Leicester.
The 24-year-old made his Rams debut as a substitute in the defeat at Watford and has started the last three matches against Crystal Palace, Cardiff and Birmingham.
"We've been very pleased with his performances so far and we would like to extend his loan," Clough told the Derby Telegraph.
"We will have a chat with Norwich and, hopefully, try and get something sorted to keep him."
The Rams boss added: "Chris has been very unlucky not to get a goal, you know. He could easily have had himself a goal or two, or even three.
"He had a goal disallowed at Cardiff that was onside and Julian Speroni pulled off a good save from him when we played Crystal Palace.
"At Birmingham on Saturday he was denied by a brilliant save and another shot struck the heel of a defender as he has turned his back. Chris is getting into the positions, which is good."