The 26-year-old forward linked up with the Rams in October on a short-term loan agreement from Tottenham Hotspur.
That spell is due to end in January, at which point he will be expected back at White Hart Lane.
Dawkins is a product of Spurs' academy system, but is yet to make a competitive appearance for the club.
McClaren hopes the lure of regular football at Pride Park will work in his favour, with transfer talks planned for the New Year.
Asked if he would like to keep Dawkins, the Rams manager told the Derby Telegraph: "Yes, absolutely. There's no doubt about that. When he came here we wanted to take him permanently.
"I have known him for a couple of years. I took him to Twente, for two weeks, and I have always kept my eye on him.
"When Johnny Russell was injured in October I thought it was a great opportunity to bring in Simon and he has impressed.
"I think he has always been a good player but to make a difference you have to have an end product, you have to score or assist.
"Simon is proving a big asset for us. He has got stronger and fitter with each game."