A foreign ministry spokeswoman said the German government had "heard about the report" but stressed that "threats are not an appropriate way to influence our policy in Syria and Iraq".
"There will be no change to our Syria and Iraq policy," she added.
A terrorism monitoring group reported on Tuesday that the extremists said they would kill one of two German hostages, who were captured from a sailing yacht in April, unless a 250 million pesos (€4.4 million) ransom is paid and Berlin stops supporting the US-led campaign against the Islamic State group in Iraq and Syria (Isis).
US-based SITE, which monitors extremists' activities worldwide, released a picture attributed to the Abu Sayyaf group showing a masked militant with a machete beside a grey-haired white man, reportedly a German doctor.
A younger white woman, said to be the doctor's girlfriend, with close-cropped hair was also shown, while other masked men stood behind them, armed with assault rifles and machine guns.
Abu Sayyaf, a small band of Islamic militants based in the southern Philippines, vowed to kill one of two hostages if the two demands were not met within 15 days, according to SITE.
The foreign ministry spokeswoman said the German government had set up a crisis task force on the case "and we will of course continue our efforts to secure a release".
The German government has ruled out joining air strikes or any ground offensive in the fight against Isis jihadists.
But it announced late last month that it would send weapons to the Peshmerga Kurdish fighters in northern Iraq to help them fend off the militants.
It has also sent around 40 soldiers to help train the Kurdish fighters in Iraq.
The move broke a taboo in Germany which, burdened with its past aggression in two world wars, generally shies away from foreign military engagements and has never before exported weapons into war zones.
The US has carried out air strikes in both Syria and Iraq, while France has begun strikes in Iraq.
Philippine authorities had previously confirmed that Abu Sayyaf abducted a German man and woman in April as they were sailing in a yacht to the Philippines from Malaysia.
However, many news organizations refrained from reporting the abductions.
Hard to judge veracity
Hard to judge veracity
Philippine authorities said on Wednesday they were investigating the reported threats and photo, but could not comment further.
"It is hard for us to make a judgement on what is said, especially on social media and the veracity of this picture, until investigators are able to determine how authentic it is," military spokesman Lieutenant-Colonel Ramon Zagala told reporters.
Abu Sayyaf, considered a "foreign terrorist organisation" by the United States, is a loose band of several hundred Islamic militants originally organized with Al-Qaeda funding in the 1990s.
It operates on remote and largely lawless southern Philippine islands that are mostly populated by the nation's Muslim minority.
Abu Sayyaf has been blamed for the Philippines' worst terrorist attacks, including kidnappings, abductions and beheadings of foreign and local hostages.
It was also blamed for the bombing of a ferry on Manila Bay that killed more than 100 people in 2004.
The group is believed to be currently holding several other foreign and Filipino hostages, including two European birdwatchers abducted in February 2012.
In recent months Abu Sayyaf has uploaded videos on the Internet proclaiming its allegiance to the Islamic State group, which has taken control of large parts of Iraq and Syria.
The US military has had a rotating force of a few hundred soldiers based in the southern Philippines since 2002 to train local troops in how to combat the group.