The deceased was among 15 European holidaymakers being held hostage for almost five months, German television reported on Tuesday.
ARD television said the woman was a mother of two and that the kidnappers had probably buried her body in the southern Algerian Sahara, where daytime temperatures have touched 50 degrees Celsius in recent months.
The German foreign ministry informed the woman's relatives of her death on Tuesday morning, ARD said.
A spokesman for the German foreign ministry said, "In the interest of the hostages who are still in a very difficult situation, the government cannot give any information. The government continues to do everything in its power to help the affected.”
News reports in recent days had said the hostages had been moved to neighbouring Mali. ARD said efforts to secure their release had redoubled since the move.
Switzerland has sent a police officer to Mali to join around a dozen Dutch and German experts working on the case there. One report, quoting Algerian security sources, said the kidnappers had accepted an army offer of safe passage to Mali in exchange for freeing the tourists.
"We are not denying nor confirming anything. We all want to get the hostages freed, all the work the state is doing is aimed at preserving the lives of the hostages," a senior official at Mali's ministry for territorial administration said.
|"The government continues to do everything in its power to help the affected.”|
--German Foreign Ministry
Officials in Mali say security forces have moved to step up patrols in the Kidal region, on the eastern side of the border with Algeria, and enlisted the help of local nomads in their search for clues as to the hostages' whereabouts.
Some 32 European tourists were kidnapped in late February and early March while travelling without guides in separate groups in southern Algeria, famous for its ancient grave sites.
Seventeen hostages were freed in May when Algerian commandos stormed a hideout in the desert, killing the kidnappers.
The authorities suspect the kidnappers are members of the Salafist Group for Preaching and Combat, who have been fighting for an Islamic state in Algeria.