A video from recent weeks in possession of the German government shows the three young children alive, an official revealed to the paper.
“The children look exhausted,” the official said.
The Saxony family of health workers Johannes and Sabine H. has not been seen since they were abducted along with four other foreigners in the country's Saada province in mid-June. Authorities there believe that the kidnappers may be Shiite rebels from the mountainous region near the Saudi border.
Three of the hostages – two German women and a South Korean woman - were found murdered by Yemeni authorities three days later.
The Foreign Ministry has refused to confirm the report that the children could still be alive, news agency DPA reported.
“The crisis squad continues to find a solution to the problem,” a spokesperson told the agency.
Yemeni officials launched a massive search for the remaining hostages, but since then there has been no trace of the family – which includes Lydia (5), Anna (3) and Simon (1).
Foreigners are often kidnapped in Yemen by tribesmen to be used as bargaining chips in disputes with the government. More than 200 foreigners have been abducted over the past 15 years.
In June 2000, a Norwegian diplomat was killed in a gunfight between police and his abductors.
In December 1998, three Britons and an Australian seized by Islamist militants were killed when security forces stormed the kidnappers' hideout. In March of this year, four South Korean visitors to Yemen were killed in an Al-Qaeda suicide bombing at a historic tourist site.