Pep Guardiola's side are currently 8/1 to land the centre-back during the January transfer window, ahead of local rivals Manchester United.
Only Liverpool, who have also been tipped to bid for the Saints star (er, Micky Quinn on Talksport), currently occupy shorter odds than City, whose centre-back and captain Vincent Kompany suffered the latest in a succession of injury blows this weekend.
The Citizens signed John Stones in a deal reportedly worth £50 million in the summer, but have often been forced to play left-back Aleksandar Kolarov alongside him in a central role.
Van Dijk, who has 11 caps for his country, has excelled since arriving at St. Mary's from Celtic 16 months ago and signed a new six-year deal earlier this year.
Should an offer from Manchester City arrive, though, it is one the 25-year-old would likely find hard to turn down.