Barca have already settled personal terms with Arthur after an infamous trip to Brazil for a club delegation a fortnight ago.
After threatening to take Barca to FIFA over their contact with the player, Gremio are now prepared to hold negotiations.
Their sports director André Zanotta declared on Wednesday: "We are ready to listen."
Arthur's deal carries a €50m buyout clause - a price regarded as too steep by the Catalans. However, Gremio are now prepared to negotiate a more realistic price next month.