Werner says Southampton have not yet tabled a bid, but the club has enquired about his availability for a summer transfer.
Goiginger scored 14 goals and assisted 17 in 40 appearances last season. That form saw him called up to the Austrian national team last November.
The Austrian has been strongly linked to play under countryman Ralph Hasenhuttl at Saints.
With Southampton in dire need of further attacking options for next season, it appears the club has singled out Goiginger as a relatively cheap addition.
According to the Echo, a bid of around £5million could force LASK Linz’s hand to sell a player as prolific as Goiginger was for them last season.