Both are due to become free agents once they reach the end of their current deals at St Andrew’s.
Clark had hoped to prevent two top talents from slipping through his clutches, with Burke a highly-rated and creative winger and Zigic a proven international striker.
Extension talks have come to nothing, though, and Birmingham have been forced to reluctantly concede that efforts must now be made to find replacements.
Clark said in the Birmingham Mail: “It has been reported we have been in negotiations with Chris Burke and Nikola Zigic. We have offered them both deals.
“Chris has obviously got numerous offers which will just blow ours away. It is very, very doubtful that Chris will come back.
“I actually spoke to him on Monday morning and I fully understand where he is coming from.
“He’s 30, still at a good age, and it is a good opportunity for him as a player – what you call a really big contract.
“I know there are lots of suitors in the Championship, probably every other club. So it is highly unlikely that Chris will come back.
“Nikola hasn’t accepted our offer, he hasn’t been in touch, so we will be looking for a replacement for him. We won’t be going forward with Nikola Zigic.
“We thank them both for their efforts and I am sure they will be remembered as very popular players in Birmingham’s history. But we have to move on now.”