The 22-year-old defender has alerted a host of top European clubs since arriving at Steaua Bucharest in January 2012, following a series of impressive performances.
Blues boss Roberto Di Matteo has been linked with Chiriches, but so have numerous other sides that are keen to bolster their ranks in January.
A regular for the Romanian national team, Milan boss Massimiliano Allegri has identified the player as the perfect successor to Thiago Silva, after the Serie A giants cashed in on the Brazilian stopper, selling to Paris Saint Germain during the summer.
Chiriches' reputation has soared over the past nine months, and was even linked with Barcelona during the summer transfer window, while Ligue 1 side Lille are also believed to be monitoring his situation.
However, Milan now appear to be the club leading the race for his in-demand signature, so Chelsea will have to move quickly if they retain any desire to ensure his future is played out in west London.