But with his love of Germany would he move to the Premier League?
It came after the holding midfielder played a starring role in the Brazil side that won the FIFA Confederations Cup in the summer. Despite his obvious class he was not at all ensured of a place in the Bayern Munich side after the arrival of Thiago Alcantara - and after a season in which he did not feature regularly.
The holding midfielder position is one Arsene Wenger wanted to strengthen in the summer - and Gustavo soon emerged as a standout option. It was even reported at one point that a deal was done - worth his £17.5 million release clause.
However that was not the case. For the same fee, his release clause with Bayern, he left for fellow Bundesliga outfit VfL Wolfsburg on the 16th August - choosing to stay in Germany, the country he had lived since joining Hoffenheim in 2007.
He explained his decision back in August, stating:
"I wanted to experience things that I no longer lived in Bayern. Bayern, you prepared the game already knowing you were sure to win the three points. I did not want to leave Germany because it is a country where I fit and I love.”
But, with Mathieu Flamini’s star fading and the fans clearly in uproar over the cumbersome performances of Mikel Arteta - could Arsenal move back in for him this summer?
It is certainly an area they must consider improving this summer - and reports suggest he has a release clause in his current deal if Wolfsburg do not get in to Europe.
At the moment they are sixth - the final European place - and just three points clear of main challengers Mainz. So it is certainly touch and go as to whether they will make Europe this season.
And with a chance ti impress at this summer’s World Cup Wolfsburg are obviously concerned - a point made clear by their general manager Klaus Allofs.
"We are very happy to have Luiz Gustavo. He is a regular first-team player and he is also an important player for the team. He has a long-term deal at our club. Of course, he is a Brazil international player and we know there could be a lot of changes to this if he is going to play a pivotal role at the World Cup."
The danger of course is that Gustavo seems to love living and playing in Germany. The likes of Borussia Dortmund, Schalke 04 and Bayer Leverkusen are sure to be trying their best to secure the funds to sign him up should this premonition of Allofs come true.
And if they do Arsenal are likely to get nowhere in their pursuit of the player.