The forward, who is the son of former Barcelona player and French international Lilian Thuram, made his mark coming off the bench for France during the FIFA World Cup.
Thuram, 25, is enjoying a great season at Borussia Mönchengladbach so far and is looking to take a step up by moving to a bigger club soon.
He is out of contract in a few months and Barcelona are always looking for good opportunities via free transfers. They apparently have been following him for a while and even saw him play live. The Catalans appreciate his ability to play either out wide on the left and through the middle, among many qualities.
However, Chelsea have one key advantage: they are willing to pay a transfer fee to get him early. Borussia Mönchengladbach would rather take some money for him in January than let him go completely free.