Gonzalez can play in central midfield as well as on the right, and has filled in at right-back for Lazio on a number of occasions.
His versatility would make him an ideal signing as he would be able to give Koeman options across at least three positions, and his ability to play at right-back could be key following Callum Chambers' departure to Arsenal.
Gonzalez has been at Lazio since 2010 and has made over 100 appearances for the Serie A side in that time, and the 29-year-old would add some vital experience to a predominantly young Southampton squad.
The club has already reinvested a large amount of the transfer money they received from the sales of their stars this summer, and the capture of Gonzalez would take their spending over £50 million.
A number of new faces played at Anfield in yesterday's Premier League opener against Liverpool, and the Saints were unlucky not to have picked up at least a point after a strong showing.
The club should be set for another successful campaign despite a major overhaul in the close season, and the arrival of Gonzalez will add more quality to the midfield.
Morgan Schneiderlin is still being strongly linked with a move away from St Mary's despite playing in the game at Anfield, and Gonzalez may be the man to replace Schneiderlin if he does depart before the end of the month.