"The deal is not yet official but it is in the last leg," Reuter told reporters in Germany on Wednesday.
The Greece international has made just two appearances in all competitions for Augsburg this season and would become Paul Lambert's first signing since taking over from Mark Hughes on Monday.
The 24-year-old is expected to offer cover to current left-back Erik Pieters as the new boss looks to bolster the leakiest defence in Europe's top five leagues.
Stafylidis, who spent the 2014-15 season on loan to Fulham, joined FC Augsburg in 2015 from Bayer Leverkusen, making 40 appearances and scoring five times for the German club.