The likes of Everton, West Ham United and Watford had all been linked with the 25-year-old, but Southampton now appear to have won the race for his signature.
ESPN reports that the Saints have come to terms on a price for the towering centre-back, whom they view as a replacement for Virgil van Dijk following his January move to Liverpool.
Vestergaard is now expected to undergo a medical on the south coast on Monday before the deal can officially be confirmed.
The Denmark international travelled to the World Cup with his country this summer but did not feature in Russia as the Danes crashed out in the last 16.
Vestergaard spent two seasons with Monchengladbach, scoring seven goals in 83 appearances.