The 31-year-old suffered the blow just 16 minutes into the Arsenal’s 3-1 victory over Bournemouth at the Emirates last Sunday.
Debuchy, who joined the Gunners from Newcastle United in 2014, was making his first appearances in 12 months, but now faces an extended spell on the sidelines.
In Friday’s press conference, manager Arsene Wenger told reporters: “It’s a severe hamstring injury, yes. Six weeks, I think. It’s frustrating for me but even more for him, he’s a guy who fights hard, he has an exceptional attitude.
“He’s absolutely disappointed, but he’s a fighter, he will fight to come back.”
The news comes one day after Santi Cazorla was ruled out for three months following surgery on an Achilles problem.