Khedira suffered torn knee ligaments in his national side's 1-1 draw with Italy on Friday and is expected to be sidelined for at least six months.
However, Gundogan, who is currently injured with a nerve injury and is not expected to return until after the Bundesliga's winter break, is hopeful he can make an impact in Khedira's absence.
"I regret very much the injury to Sami. He is not only a colleague, but also a good friend of mine. It’s very sad news,” Gundogan told Bild.
Talking of his own injury woes, the 23-year-old said: "Slowly but surely it is getting better. I've even been doing sprint training.
"Of course, I hope it goes as quickly as possible. But still I am not completely pain free, so I still need treatment.
"It's quite possible that I can get fully fit after the winter break. Of course I want to get back as quickly as possible in the Dortmund team. But Brazil is top priority."