The 26-year-old has refused to commit his future to the North London club should they fail to qualify for next season's Champions League.
City are said to be looking for a centre-back to partner Vincent Kompany, and they have identified the Spurs man as their main target this summer.
Vertonghen has been recommended to the City hierarchy by Belgium international teammate Kompany, according to Goal.com.
The defender has previously been linked with a move to Barcelona, but it is believed that the La Liga club are no longer interested in securing his services.
Vertonghen joined Spurs from Ajax in 2012.