The 19-year-old fullback broke into the Lille first team last season, and Manchester United and Chelsea dispatched scouts to monitor his progress during last week's Lille versus Rennes match.
Digne has represented France at every youth level from under-16's up to under 19, and has impressed for Lille so far this season.
Chelsea are thought to be considering their options at fullback, with current left-back Ashley Cole out of contract at the end of the season.
Blues boss Roberto Di Matteo is believed to remain hopeful that a deal can be agreed, and Digne could act as a long-term understudy for the 31-year-old Cole, or could replace him at Stamford Bridge if the England international departs in the summer.
At Old Trafford, Sir Alex Ferguson is reportedly looking for a replacement for French international Patrice Evra.
Manchester United signed Dutch fullback Alexander Buttner over the summer, but Ferguson is thought to want several options in place with a view to succeeding Evra as Manchester United's long-term left-back.
Digne extended his Lille contract until 2016 earlier this year, and is highly-rated by the Ligue 1 side after graduating through their academy.
And having sold Eden Hazard and Gervinho in the past two seasons, Lille will be keen to hold onto their latest teenage starlet.