The Hammers are still looking to strengthen the centre of their defence after a proposed transfer for Harry Maguire fell through.
They have seemingly identified Mavropanos as a potential alternative to Maguire and are now working to get a deal over the line.
According to German publication Bild, the Greece international has already agreed personal terms with the Hammers, leaving it up to the two clubs to discuss a fee.
With Mavropanos under contract until the summer of 2025, the Bundesliga club have set the player's asking price at €20m (£17m).
The 25-year-old, who joined Stuttgart from Arsenal in 2022, has made 89 competitive appearances for the German side, chipping in with eight goals.
Over in West London, Fulham are reportedly interested in signing Ivory Coast international Eric Bailly from Manchester United.
Bailly is expected to leave Old Trafford this summer after being deemed surplus to requirements by Man United boss Erik ten Hag.
Al-Nassr were believed to be showing an interest but have seemingly turned their attention to Manchester City centre-back Aymeric Laporte.
According to the Daily Mail, Fulham have asked Man United about the 29-year-old with the Cottagers looking to make further defensive additions after recruiting Calvin Bassey from Ajax earlier this summer.
Marco Silva's side could face competition from Turkish giants Galatasaray, who have been linked with a move for Bailly.
With Man United keen to offload the central defender, they are supposedly willing to terminate his contract to aid his search for a new club.
The Ivorian spent the 2022-23 season on loan at Marseille, where he featured in 23 matches across all competitions.
Elsewhere, Saudi Arabian club Al-Hilal have reportedly reached an agreement to sign Moroccan goalkeeper Bono from Sevilla.
After recently announcing the signing of Neymar from Paris Saint-Germain, Al-Hilal are believed to be on the verge of unveiling their latest summer addition.
According to journalist Fabrizio Romano, the Saudi Pro League side have agreed a €21m (£17.9m) deal with Sevilla for the 32-year-old.
A medical has been booked for the Morocco international, who will put pen to paper on a three-year deal with Jorge Jesus's side.
The shot-stopper is set to leave Sevilla after making 142 competitive appearances and winning the Europa League on two occasions.
Bono was in goal for Wednesday's UEFA Super Cup against Manchester City, but he could not prevent the Citizens from winning the subsequent penalty shootout after the contest had finished in a 1-1 draw.
Meanwhile, on the South Coast, Bournemouth's move for Leeds United midfielder Tyler Adams has reportedly been thrown into doubt due to a dispute over the player's release clause.
A recent report claimed that the Cherries had triggered the player's release clause, which would have allowed them to discuss personal terms with the 24-year-old.
However, according to The Athletic, the proposed transfer appears to be in doubt after Leeds claimed that the time-limited relegation clause was not activated.
While Bournemouth claim to have submitted the required £20m bid on Monday, Leeds maintain that the clause was not triggered and has now expired.
The USA international has experienced an eventful couple of weeks after a proposed move to Chelsea recently fell through.
Adams, who joined Leeds from RB Leipzig last summer, is currently sidelined with an ankle injury and is expected to return to action after the international break.