The 19-year-old will allegedly leave BVB at the end of the 2019-20 campaign, and the likes of Manchester United and Liverpool have been strongly linked with the England international.
However, according to France Football, Chelsea have stolen a march on their Premier League rivals by holding talks with the teenager's representatives.
Sancho has again been in terrific form for Lucien Favre's side during the current season, scoring 16 times and registering 17 assists in 30 appearances in all competitions.
The Englishman, who turned down the chance to sign a new deal with Manchester City to move to Germany, is thought to be leaning towards a switch to Old Trafford.
Barcelona, Bayern Munich and Real Madrid have also previously been credited with an interest in the attacker, who could fetch upwards of £120m during this summer's transfer window.