Barca president Sandro Rosell made the verbal offer to Arsenal chief executive Ivan Gazidis.
But the bid from the European champions was immediately rejected. Barca - having ended the phoney war - are now ready to increase their bid to £30m.
But that will not be enough to tempt Emirates manager Arsene Wenger to sell.
However, a £35m offer could seriously test Arsenal's resolve to hang on to Fabregas, 24.