After seeing an initial £15 million offer turned down last week, the Gunners had hoped that an improved bid for the 22 year-old would be accepted.
Their second bid totalled £25 million, although it was reportedly rejected by the Scottish champions as it included too many add-ons, according to the 'Daily Mail'.
Celtic are believed to be looking for the £25 million up front in what would be a Scottish-record fee, and Unai Emery's side are expected to make another offer as they look to sign a first-team player before the new season starts.
Arsenal have made just one signing so far this summer, bringing in young striker Gabriel Martinelli from Brazilian side Ituano.