Barcelona agreed a purchase option for the 21 year-old in March, giving the club the option to sign the player for a fee that could reach €39million.
The departure of Andres Iniesta and an injury to Carles Alena has left Barca manager Ernesto Valverde facing a potential shortage of midfielders and the club is understood to be hoping to sign Arthur earlier than originally agreed.
Bolzan Jr denied any chance of the move being brought forward, telling Sport: "The position of Gremio is the contract that we signed.
"There is no new factor that could mean that we have to renegotiate with Barcelona or that we can undo what we have already signed.
"What we agreed is that Barcelona has the right to carry out the exercise of preference, but once it is executed the player will only go in January."
Arthur, who scored three goals in seven appearances for Gremio in the 2018 Campeonato Gaucho, has twice been an unused substitute for Brazil after playing for his country at under-17 level.
Bolzan Jr insisted his stance on the player's future is clear and said he would not be swayed by speculation from Spain.
"A lot of news comes out of Gremio, a lot of things come from Spain, and every moment someone points out something about Barcelona doing this or that it generates a climate of doubt," he said.
"But Gremio has a clear and defined position and there is no new situation that changes it."