He did not feature at the club, primarily due to the form of Simon Kjaer and Andreas Christensen for their clubs last season, Vestergaard still managed to attract the interest of four Premier League clubs.
He is now due for a medical on the south coast of England on Monday, and will become Southampton's third summer signing, following the arrivals of Mohamed Elyounoussi and Stuart Armstrong.
Sky Sports have provided further information on the deal, reporting that Southampton have agreed a fee of £18m for the Danish defender.
They also back up the report from Kicker, saying that Vestergaard will have a medical with the club 'this week'. Furthermore, it looks like the transfer could go through rather quickly, as personal terms are 'not expected to be a problem' in negotiations between Southampton and the player's representatives.
Southampton have been in the market for a central defender since they sold Virgil van Dijk to Liverpool in January for a world-record £75m fee.
With Maya Yoshida, Wesley Hoedt and Jan Bednarek as the only recognised centre-backs in the Southampton squad, Vestergaard would be a welcome addition to Mark Hughes' squad.