Clough has been an admirer of Burke for some time and monitored the 29-year-old's contract situation at St Andrew's this summer before the Blues took up their one-year option on the Scotland international.
Jeff Hendrick's injury, the Rams academy product is out for three months with an ankle problem, has left Clough short of options on the right-hand side and he has already drawn up a list of potential loan targets - which includes Burke.
"Chris is one we have talked about, certainly," said Clough.
"But it depends on player availability and all sorts of other things. We have already made some enquiries this week so we have a couple of people thinking about things and a couple of clubs have said they want to hang on and see what happens.
"It is a bit of a waiting game but very rarely in the loan market do you go for your number one target and everything slots into place, so you have to be patient."
Burke made his first league start of the season prior to the international break as Birmingham drew 1-1 with Ipswich, scoring the Blues' goal against the Tractor Boys.
He has also started two Capital One Cup matches and made three substitute appearances in the Sky Bet Championship during the current campaign.