Elmander, 29, is ready for a new challenge in Turkey with Galatasaray when his contract ends next month.
The Sweden striker is expected to play his last game for the club against Manchester City after scoring 12 goals in all competitions this season.
High on Coyle's list is a ready-made replacement and the Glasgow-born boss had made no secret of his admiration for Daniel Sturridge.
The 21-year-old has weighed in with eight goals in 12 appearances after arriving on from loan in January.
Coyle said: "I would love to have players of that stature at the club and I will leave no stone unturned to do that. We all recognise how finance can help you.
"If Johan goes that will free up a lot of finance because he was our record signing and on a terrific contract.
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