Sport says after marathon negotiations, an agreement was close to being finalised yesterday morning between the two clubs.
Barca chiefs Andre Cury and Robert Fernandez are in Porto Alegre this week and thrashed out terms at Arena Gremio with high-ranking officials from the Brazilian club, including president Romildo Bozan.
Arthur's buyout clause is set at €50m, but Barca's offer of €30m plus €10m in bonuses has impressed Gremio. The board met in the afternoon on Monday to consider the offer and it was looked upon favourably.
The two clubs, should the final negotiations run smoothly, could make an announcement later this week.