Sky Sports sources understand that Arsenal have offered Bayern £17.2million for Gustavo as Arsene Wenger seeks his first major summer signing.
Gustavo's future at Bayern is uncertain after the capture of Thiago Alcantara further reduced his prospects of regular first-team football with the European champions.
He has also been linked with the likes of Napoli and Wolfsburg, and the 26-year-old wants to be playing regularly to boost his prospects of representing Brazil at next summer's World Cup.
"My contract with Bayern runs until 2015 but it is very important that I play first-team football to keep getting called up for Brazil," Gustavo told reporters on his arrival in Basel for Brazil's friendly against Switzerland.
"I am in a very delicate situation at Bayern and this needs sorting out. I have personal and professional reasons to move.
"I have heard about the Arsenal interest. It's a big club we're talking about here so of course I'd like to play for them.
"Arsenal play a very attractive brand of football and are in the Champions League. They also have a very clever manager in Arsene Wenger.
"Even when Bayern beat them in the last Champions League [on aggregate] you could see they are not pushovers."
Arsenal's talks to sign Gustavo are understood to be at an advanced stage, although reports in Germany have also claimed that clubs in the Bundesliga, Spain and Russia are interested.
Gustavo started just 16 Bundesliga games for treble-winning Bayern last season and it is understood he would prefer to move to a team in the Champions League.
And, as Arsenal await a marquee signing which may or may not materialise before the September 2 closure of the transfer window, 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, who has made two bids for Liverpool striker Luis Suarez, 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 on Saturday.