Derby boss Nigel Clough this week revealed the Scotland international remains one of several targets as he looks to sign a right-sided midfielder, stating: "Chris is one we have talked about, certainly".
It comes after Leeds showed interest in Burke late on in the transfer window, following a summer in which Birmingham activated a contract clause which keeps the 29-year-old at St Andrew's for another year.
And Clark insists Burke, who has played a bit-part role for Blues so far this season, remains a vital part of his plans amid the latest loan rumours.
"From my point of view, and I can only speak as a football manager, Chris has never indicated to me that he wants to move, I've never indicated to Chris that I'd like him to move," Clark said.
"He's bright, he seems to be enjoying his football and since I've become manager not once has Chris or anybody involved with Chris asked for him to leave.
"Something like that doesn't make any sense to me - letting one of your best players go out on loan. If it was before the window shuts and, for the well-documented reasons, he moves because he goes for a huge transfer fee and the saving of his salary then maybe, not that I would want that to happen.
"But, certainly, a loan move, I don't see what benefit at all that is to anybody."
Clark added: "I would say most of the Championship are interested in him because he's a top-quality player.
"I told everyone before the deadline day about a conversation I had with another Championship manager who thought he was putting a substantial offer to me verbally. And I went back and politely said that wasn't acceptable, they didn't come back to the table.
"I'd imagine Nigel (Clough) would be in a queue of a fair few managers interested in Chris Burke."
However, one player who could well be set for a loan move away from the Championship club in order to build up his match fitness is Will Packwood.
The American full-back, 20, has not featured since suffering a double leg break in early January and may find himself loaned out to get back up to speed having now returned to full fitness.
"I might, I'm not saying this is for definite, I might consider a loan move for him to a League One or Two club to just accelerate his recovery," Clark said.
"It would just be for our benefit, not sending him out on loan because he's surplus to requirements, far from it."