The 25-year-old, who took in a spell with Aston Villa last season, has been training with the Rams in an effort to earn a deal.
Former Reading and Bristol City midfielder Kalifa Cisse is another trialist to have been working with Derby of late, as he attempts to find a new club.
McClaren has been impressed by both men and is determined to get deals done after seeing his first-team squad depleted by injuries.
He said in the Derby Telegraph: "I know Simon very well. I very nearly took him to Holland with me when I was at FC Twente and he was at Queens Park Rangers for a couple of weeks when I was there. I know what he can do.
"He has come in and adjusted and there might be a possibility of doing something with him. We are looking at that."
Pressed on whether a deal could be done in time for Dawkins to face Watford on Saturday, McClaren said: "Maybe. We will discuss that. We'll wait and see, knowing also we have got a few injuries and we are a little short of numbers in the squad for Saturday."
On Cisse, who is available as a free agent, McClaren added: "Kalifa has good pedigree and good history with Reading and Bristol City in the Championship and the Premier League.
"We have watched games of him at Reading, so we know what he can do. It has been a good opportunity to get him and Simon in and training with us.
"They have done well and the boys have been tremendous with them. We are one big family and they have come into the family well.
"They have been training three days and now we will sit down and discuss it further."