The 23-year-old arrived at the Emirates from Greek side PAS Giannina in January 2018 but has struggled to force his way into Mikel Arteta's plans since the Spaniard took charge.
Mavropanos is in the second year of his loan spell at Stuttgart after impressing in the Bundesliga last season, and the deal is understood to include an option to buy.
According to Bild, Stuttgart director Sven Mislintat - formerly head of recruitment at Arsenal - is keen for the player's £2.6m purchase clause to be triggered so long as his side avoid relegation to the 2. Bundesliga.
The report adds that Mislintat remains convinced of Mavropanos's quality after previously bringing him to Arsenal, but the Greek's injury woes may still prevent a stumbling block.
Mavropanos already has two goals from four appearances across all tournaments for Stuttgart this season, and his deal at Arsenal is due to expire in 2023.
CSKA Moscow 
FC Rostov 
Leicester 
Chelsea 
Valencia 
Betis 
Dynamo Kyiv 
Chornomorets Odesa 
Verona 
Inter 
Bayer Leverkusen 
FC Heidenheim 
Borussia Dortmund 
SC Freiburg 
VFL Wolfsburg 
Union Berlin 
Arsenal 
Nottingham Forest 
Everton 
Brentford 
Spartak Moscow 
Lokomotiv Moscow 
Rangers 
Dundee Utd 
Atletico Madrid 
Alaves 
Lens 
Marseille 
Ingulets Petrove 
Shakhtar Donetsk 
AC Milan 
Juventus 
ST Malo 
Bordeaux 
Manchester City 
Tottenham 
Eintracht Frankfurt 
Werder Bremen 
Girona 
Espanyol 
Feyenoord 
Heerenveen 
ST Etienne 
Montpellier 
Grenoble Foot 38 
Heart Of Midlothian 
Celtic 
Reims 
Lyon 
PSV 
Groningen 
Celta Vigo 
Barcelona 
Benfica 
Estrela Da Amadora 
Sparta Rotterdam 
AZ Alkmaar 
Southampton 
Liverpool