The 21-year-old is widely regarded as one of the most talented midfielders in Brazilian football, and Barcelona have long been credited with an interest.
Gremio were initially said to be unwilling to sell Arthur for anything under his £44m release clause, but according to Sport, there has been a breakthrough in negotiations, with Barcelona agreeing a £26m deal.
The report claims that Arthur will officially join Barcelona ahead of the 2018 World Cup, where it is believed that the midfielder has an outside chance of making the Brazil squad.
That said, it is thought that Arthur could then re-join Gremio on loan until the end of the Brazilian season in December.
The midfielder's rights are currently part-owned by Gremio, his agent and an outside investor.