Aged 26 the Canadian International is out of contract at the Loftus Park club but had originally insisted that he would be staying at the Hoops.
Neil Warnock is already on a tight budget at Bramall Lane and has very little to spend if anything. However he is currently in talks with Clarke Carlisle who is also a free transfer in the summer at QPR.
Bircham told the Sheffield Star: "I'm seeing what might be available out there. It's the sensible thing to do in the circumstances.
"I've got no wish to leave QPR but it would be wrong not to see what might be out there if I have to."