The 25-year-old made 32 league appearances for Monchengladbach last season as they finished ninth in the Bundesliga table.
A number of Premier League clubs had been linked with Vestergaard, but Saints appear to have won the race for his signature.
According to Sky Sports News, the centre-back is now set to finalise personal terms and sign off the relevant paperwork.
The report also claims that Southampton are hopeful of having the £18m transfer completed by the start of next week.
Vestergaard has spent two seasons with Monchengladbach, scoring seven goals in 83 appearances.