The 25-year-old who went to the World Cup in Russia to play for Denmark but didn’t play a game, will undergo a medical this week for the Saints.
The report claims that Southampton won’t have any problems to agree personal terms with the player, who could become Saints’ third signing of the summer following the arrivals of Mohamed Elyounoussi and Stuart Armstrong.
The Saints are looking to bolster their defensive options since selling Virgil van Dijk to Liverpool last January, and it seems they have found the right candidate for the role.
The Danish defender is highly regarded in Germany after spending the past eight seasons there with Hoffenheim, Werder Bremen and Monchengladbach.
Several Premier League clubs, including Everton, West Ham United and Watford, have been linked with Vestergaard, but as it stands Southampton have won the race for the towering defender.