The Germany international - a £42million buy from Real Madrid last summer - suffered the problem during Tuesday's 1-1 last-16 draw at Bayern Munich which saw Arsenal exit the Champions League.
Speaking ahead of Sunday's north London derby with Tottenham, manager Arsene Wenger told arsenal.com: "He's out. I don't know for how long but it is a hamstring injury. There are scans going on."
In a further blow, given Jack Wilshere's absence with a broken foot, midfielder Aaron Ramsey's return from a thigh injury has been delayed.
Wenger added: "We have had to defer his comeback to full training so he is not available. He is having difficulties in the final stages of his rehab. It looks to be short term."
Left-backs Kieran Gibbs and Nacho Monreal have a "good chance" of being available. Gibbs has a fitness test on Friday.
Swedish midfielder Kim Kallstrom could be in contention for his debut after signing despite a back injury, while Yaya Sanogo is available.
Wenger said: "Kallstrom joins the normal training group today and Sanogo too.
"I will have to decide if Kallstrom is okay to be in the squad on Sunday. I will see how he looks. It is not impossible."