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The Latics entered the bidding not too long ago with an initial approach of £500,000, before raising this to £600,000.
This third bid is also unlikely to be the last we hear of the transfer with Latics boss Paul Jewell seemingly determined to get his man.
"We have had another offer for Robbie, which myself and the chairman have spoken about and turned down," said Burnley chief executive Dave Edmundson.