Sport claims that the technical secretary of the La Liga giants, Robert Fernandez, recently had a personal meeting with Arthur's agent to discuss the midfielder's potential move to the Nou Camp.
Fernandez travelled to South America earlier this month to scout potential targets after Barcelona president Josep Maria Bartomeu confirmed that the club will try to improve Ernesto Valverde's squad during the upcoming January transfer window.
"We are happy with the squad that we have but if things can be done to improve then they will be done, because we always aspire to more," Bartomeu recently admitted.
Several reports in Spain in recent days have claimed that 21-year-old midfielder Arthur was one of the main reasons behind Fernandez's trip.
Barcelona, nevertheless, remain on the hunt for a new playmaker to become the long-term replacement for 33-year-old Andres Iniesta after they were unable to sign both Marco Verratti from Paris Saint-Germain and Philippe Coutinho from Liverpool.
Fernandez has therefore been watching both legs of the Copa Libertadores finals between Gremio and Lanus.
Arthur's side claimed the title for the first time in 22 years after securing a 1-0 advantage at home before a 2-1 win in Argentina on Wednesday night (29 December).
Sport claims that, following the first leg played at Porto Alegre on 22 November, Fernandez had a dinner with Arthur's agents in the Brazilian city to sound out whether the player would consider a move to Barcelona.
The Spanish publication suggests that the response was positive with Arthur willing to join Barcelona despite having attracted the interest of other big European clubs likes Manchester United.
Sport points out that Brazilian media are claiming that Jose Mourinho is also pressing to lure him to Old Trafford while earlier this month Daily Mail reported that Chelsea and Atletico Madrid were also monitoring his development.
The Spanish publication understands that Barcelona are the frontrunners in the race due to Arthur's preference to move to the Nou Camp, but the Catalans will need to act quickly if they really want to win the race.
Nevertheless, Sport predicts that United won't be the only competitor after Arthur delivered a Man of the Match performance during the second leg of the Copa Libertadores.
The young midfielder helped Gremio secure a 2-0 advantage in that game before being forced off after 50 minutes with an ankle injury.
Barcelona are meanwhile expected to make a new attempt to sign Coutinho from Liverpool. However, Arthur plays in a dipper role and luring him away from Gremio would be much easier as he has a €50m (£44m, $59m) release clause in his contract.