Giroud is under consideration at BVB for the January market and there's now plans to make a firm approach to the Gunners.
BVB see the centre-forward as a replacement for Pierre-Emerick Aubameyang, who is expected to be sold next month after angering management.
Le 10 Sport says the prime target to succeed Aubameyang is Giroud, knowing the Frenchman wants to leave Arsenal for regular first team football and to strengthen his World Cup prospects.
Giroud said only this past weekend: "In January, I will have to ask myself the question. What the coach advises, it matters.
"A loan can be a solution, but I do not know if Arsenal will let that happen."