HITC Sport understands that the Saints are preparing to talk terms amid reports of interest from elsewhere.
The Dutchman has impressed hugely at St Mary's since arriving last summer on a three-year contract.
He has been heavily linked - by the Sun - with a possible role with Holland, who are in danger of not qualifying for next summer's European Championship under their new boss Danny Blind - who has only just taken charge but already finds himself under pressure.
Koeman, who has extensive managerial experience with the likes of Feyenoord, PSV Eindhoven, Valencia and Benfica, is understood to be very happy with life at St Mary's.
And whilst there is no sign of him showing itchy feet at this point, Southampton are ready to open talks about extending his stay - given the excellent job he has overseen since his arrival.
The 52-year-old has seen a host of players depart, including Adam Lallana, Dejan Lovren, Luke Shaw, Morgan Schneiderlin and Nathaniel Clyne, but he has managed to build on their recent successes with a number of his own astute captures.
If offered, it is understood that Koeman would be open to the idea of penning fresh terms with the Saints.