Dzeko scored 26 goals in 29 league starts last season to fire Wolfsburg to their first Bundesliga title in history, and speculation has persisted over his future ever since.
The 24-year-old signed a new contract last summer, rejecting overtures from Serie A giants AC Milan, but Wolfsburg have struggled on the pitch this season.
Dzeko has still managed to score 18 goals, however, and there is reported interest in him from the likes of Chelsea and Manchester City. The striker has said he would be keen to join a Premier League side at the end of the season and Kostner admits he is unsure whether he will stay at the Volkswagen Arena.
"Edin Dzeko will finish the season a Wolfsburg player, that's for definite, whatever happens after that is down to rumours," said Kostner.
"Edin Dzeko saying he will stay would be an absolutely huge sign for us. I can't tell you how important that would be for us, but that's neither here nor there. Right now we have to wait to see what happens in May.
"As his manager I can't have much influence on whether he stays or not. All I can say at the moment is that he is on absolute top form. He's really firing on all cylinders. He's doing fantastic."
Dzeko will line up alongside his Brazilian counterpart Grafite against the Cottagers on Thursday as they look to overturn a 2-1 deficit from last week's first leg at Craven Cottage.
Fulham had looked to be coasting towards their first Europa League semi-final after opening the scoring courtesy of goals from Bobby Zamora and Damien Duff.
Alexander Madlung's last minute header gave the German side a vital away goal though and Kostner has called on his team to give their all to set up a last four tie with either Hamburg or Standard Liege.
"Before the game in Fulham I said that we'd need two excellent performances to get through," the 58-year-old said.
"We didn't give them two weak goals in the first leg but we made life easier for them by not concentrating for the two goals.
"Madlung's goal gives us a good chance now but what we need to do is to be determined. We have to have the will, be determined to give an absolutely excellent performance.
"We need everyone in this club, including the fans, to give a performance for 90 minutes, maybe longer and that's the way we'll get through to the semi finals."
Kostner's side warmed up for the game with an emphatic 4-0 win over Hoffenheim to move up to ninth in the Bundesliga.
The Wolfsburg manager has been boosted by the news that captain Josue and key defender Madlung should be fit for Thursday's clash after overcoming hamstring injuries.
Fulham's 2-1 win over Wigan on Sunday was their first since February but Kostner has warned against taking the Londoners too lightly.
"We can't go hell for leather tomorrow," said Kostner.
"Fulham are a compact dangerous team who are strong in defence and have big guys up front so that provides us with a dangerous situation."
Arguably Fulham's most famous fan, Hugh Grant, is reported to be attending Thursday night's match but Wolfsburg midfielder Zvjezdan Misimovic has warned the actor that his trip to Lower Saxony may turn out to be an unhappy one.
"I've only just heard that he's a Fulham fan," said the 27-year-old, who set up Madlung's crucial goal last week.
"Of course he's welcome to come to the game but come the final whistle he won't have much to laugh about."