The Spanish publication claims that officials from both clubs met on Monday evening (19 February) to iron out the terms of a deal which could rise to €40m (£35.2m, $49.3m) with add-ons.
Sport says that sources from Gremio have confirmed that Barcelona's offer is worth €30m (£26.4m) but also includes €10m on add-ons to be paid in two different installments.
The Catalans would pay the first €5m when Arthur plays a certain amount of games for Barca while Gremio would receive the remaining €5m if the 21-year-old midfielder signs a contract renewal with the La Liga giants in the future.
The news from Sport comes after earlier this week Mundo Deportivo reported that Barcelona technical secretary Robert Fernandez arrived in Brazil on Saturday [17 February] to step up negotiations to sign the promising midfielder.
Mundo Deportivo had previously reported that the La Liga giants had already reached to sign Arthur and for 21-year-old midfielder to move to the Nou Camp during the 2019 January transfer window.
And Sport now adds that on Monday night Fernandez and Barcelona representative in Brazil Andre Cury took a decisive step to completing the signing after striking an agreement in principle with Gremio officials.
The Catalan publications claims that Fernandez and Cury are expected to have a new meeting with Gremio officials on Tuesday to iron out the final terms of the deal.
It looks like the financial details are already agreed but the clubs have now to define whether Barcelona allow Arthur to remain at Gremio until 2019 as the Brazilian outfit previously demanded.
Last week, Gremio president Romildo Bolzan Junior claimed that he was unaware whether Arthur had already agreed personal terms with Barcelona but suggested the clubs had made big progress.
"We were not told Arthur was sold. I will take into account that it is a journalistic matter that may have come from internal sources. But there are negotiations and there are advances in some parts of the negotiation," Bolzan said as quoted by Globo Esporte.
"One of them is that Arthur will stay with us for a while but Gremio have not finalised and does not have a definite proposal. I will leave [this] as press speculation, not a business situation. Grêmio have not completed deal with Barcelona."
The Catalans still have to wait for Phillipe Coutinho to get the Portuguese nationality to make a space for Arthur as Ernesto Valverde already has the maximum three non-EU players in his squad, made up the former Liverpool star, Brazilian Paulinho and Yerry Mina.