Talks are at an advanced stage despite the fact the midfielder, who was part of Brazil's Confederations Cup squad this summer, is away on international duty.
Gustavo started just 16 Bundesliga games for treble-winning Bayern last season and has seen his chances of first-team football hit further by the signing of Thiago Alcantara.
Wolfsburg were close to signing the 26-year-old, but it is understood he would prefer to move to a team in the Champions League.
Arsenal face competition for the player, and reports in Germany suggest he is still weighing up his options.
Abendzeitung Muenchen quote Bayern's CEO Karl-Heinz Rummenigge as saying: "Luiz has offers from England, Spain, Russia and Germany. The player has not made a decision yet."
As Arsenal await a marquee signing which may or may not materialise before the September 2 closure of the transfer window, Arsene Wenger believes it is important to look at the players he does have available for the start of the season.
"We are in the transfer market and we try still to strengthen our squad," Wenger told Arsenal Player.
"It's important as well now to focus on the squad who are already at the club and that you focus on the competition, not only on the transfer market."
Arsenal start their Premier League campaign with a home game against Aston Villa.