The Blues boss was hoping to hold talks with the Danish international winger over the weekend after Blues undisclosed bid for the player was accepted by Chelsea at the end of last week.
Gronkjaer is currently on international duty in Portugal and Bruce is out their too, combining a holiday with work as a Euro 2004 tv pundit for the BBC.
Bruce is hoping to try and persuade Gronkjaer on staying in the Premiership with a move to St Andrew's but, understandably, with the Danes still involved in the Euro 2004 finals, permission for the Blues boss to talk to the player were denied until his countries participation in the competition is over.
Gronkjaer's former manager at Chelsea, Claudio Ranieri, is also believed to be interested in taking him to his new club Valencia.