The 18-year-old has established himself as a regular in Ajax's first team over the past two seasons, making 34 appearances in all competitions during the campaign just finished, including becoming the youngest captain in Dutch football history.
The Guardian reports that City manager Pep Guardiola sees De Ligt as a player who could fit into his philosophy at the club.
City are on the lookout for a replacement for Kompany, who has suffered from numerous injury problems in recent seasons and is entering the final year of his contract at the Etihad Stadium.
The 32-year-old's form during the 2018-19 season is expected to determine whether City will offer him an extension.
Guardiola will now weigh up how much game time he would be able to give to De Ligt before deciding whether or not to make a move for the Netherlands international, with Kompany, Nicolas Otamendi, John Stones and Aymeric Laporte already available to him.