Supposedly the offer is a slight improvement on his current terms at Atalanta, but it isn't something that has completely turned his head.
The south coast club really struggled in front of a goal last term, with neither one of Shane Long, Charlie Austin, Manolo Gabbiadini or Guido Carrillo showing any real consistency.
Carrillo in particular found the going tough in the Premier League following his January move from Monaco - the 27-year-old is yet to get off the mark and has just been left out of the club's pre-season tour of China amid speculation of a move away.
Petagna found the back of the net on six occasions last season which is slightly worrying on the face of it for Saints fans, but the Italian target man has been attracting interest from around Europe for his effective link up play.
Should Southampton beat SPAL to Petagna, then they will have a similar player on their hands to when Graziano Pelle played at St. Mary's before his move to the Chinese Super League in 2016.