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Inter Milan is also interested in the Danish 25-year-old. The club, who currently sit third in the Serie A, is looking at the option of loaning Vestergaard until the end of the season before deciding to make a more permanent move.
Vestergaard started his youth career at Danish side Brondby IF before being brought to Germany by TSG 1899 Hoffenheim in 2010. After a two-year stint with Werder Bremen, the defender made a move in 2016 to Gladbach.
The Foals currently find themselves sixth in the Bundesliga with 28 points, the same amount of points as Red Bull Leipzig, Bayer Leverkusen and Borussia Dortmund above them.
So far Vestergaard has started in all 17 Bundesliga games for Gladbach this season, although the side had its last clean sheet across all competitions on October 15 when they beat Werder Bremen 2-0.