The London club is prioritizing an improvement to Jose Mourinho's defensive options ahead of the new season.
Despite Mourinho's job being under some threat, the Daily Express reports he is already planning for next term.
He is targeting a centre-half, and given the club may be priced out of a move for Milan Skriniar from Inter Milan, they are looking at alternatives.
The one who has emerged as a key target is Verstergaard, who would cost around £18 million.
The Danish international has been with Southampton since 2018 when he arrived from Borussia Monchengladbach and has played 70 games for the club.