Scottish international Burke, 29, was the subject of late interest from Leeds United on deadline day, Blues boss Lee Clark confirmed, and now both the Yorkshire club and Derby are looking at signing Burke on loan, according to twentyfour7 Football.
"I had a conversation with Brian (McDermott, Leeds manager) on Sunday. He basically asked about Chris," said Clark.
"[Leeds] said they were willing to do that deal, but thankfully Brian didn't come back and pursue it.
"Chris is 100% happy and he's enjoying it here. He's never indicated to me that he has ever wanted to leave."
Derby manager Nigel Clough recently confirmed that they were looking to sign a right-sided midfielder on loan.