The 18-year-old only joined Lille from Paris Saint-Germain in the summer but has already broken into their first team, making seven appearances in all competitions this term.
However, the Ligue 1 side are believed to be struggling financially at present and may be forced to sell some of their squad this window to evade sanctions from the French football federation.
According to the Daily Mail, Soumare first caught the eye of City scouts for PSG's Under-19s in the UEFA Youth League.
The report goes on to claim that scouts were dispatched by the Etihad side to watch the box-to-box midfielder in action in Saturday evening's 1-0 win at Caen.
Soumare is believed to be viewed by City as a long-term replacement for Yaya Toure, whose contract expires at the end of the season.