Meyer will leave Schalke on a free transfer this summer after failing to agree terms on an extension to his deal, with the German outfit suspending Meyer for the final weeks of the season.
Bild reports that relegation-threatened Southampton were offered the chance to sign Meyer for €10m (£8.8m) but were not interested in a deal for the Germany international.
Meyer has been heavily linked with a move to either Arsenal or Liverpool when his contract expires, while Barcelona and Bayern Munich have also scouted the player and are understood to be monitoring the situation.
The 22-year-old joined Schalke in 2009 before coming through the ranks and going on to make almost 200 senior appearances for the club.
However, his exit has turned sour after Meyer accused sporting director Christian Heidel of trying to bully him into signing a new deal, leading to his suspension.