"We are facing a case that is very delicate and very difficult to resolve," Burkhalter told Le Matin Dimanche.
Beatrice Stockly, who was previously abducted by Islamists from Timbuktu in 2012, was taken for the second time on January 7th by armed men who stormed her home in the country's fabled city.
The kidnapping was claimed by Al-Qaeda's north Africa affiliate (AQIM) who issued an eight-minute video in which it denounced Stockly as "a Swiss nun who declared war against Islam".
The footage was replete with montages of jihadists brandishing weapons and the masked speaker spoke with a British accent.
Stockly, who is in her 40s, has lived in Timbuktu for years and has previously been identified as a missionary and social worker.
The last time she was snatched in April 2012, she was described as the last Westerner living in the legendary desert city who had refused to leave despite it being captured by Islamist Ansar Dine rebels on April 1st.
Two weeks later, special forces from Burkina Faso swept into rebel-held northern Mali aboard a helicopter and whisked her to safety in a pre-arranged handover by Islamist rebels.
Battling for her life
In Sunday's interview, Burkhalter said that the fact that Stockly's kidnapping was her second considerably complicated efforts to free her.
He pointed out that the last time she was kidnapped, her abductors had demanded that she leave and never return to Mali.
"That's where the difficulty lies," he said.
"We are in a battle to try to keep her alive.
"Unfortunately, that is not going to be easy."
Stockly's decision to remain in Mali has been harshly criticized, with some politicians calling for her to be handed for her rescue — if she makes it out.
Burkhalter acknowledged Swiss authorities had warned Stockly a number of times about the dangers of remaining in Mali, but said he would not judge her for ignoring the warnings.
"She is a missionary who felt her life and her role was over there," he said.
Switzerland has a clear policy against paying ransoms, he said, stressing it would only lay "the groundwork for future kidnappings . . . it would be a vicious circle."
Stockly's capture is believed to be the first in northern Mali since the kidnap and murder of two French radio journalists in November 2013.
The two were snatched from the northeastern town of Kidal and their bodies found several hours later.
Earlier this month, an elderly Australian doctor and his wife were seized by jihadists in a remote town in neighbouring Burkina Faso on the night of January 15th-16th.
Story continues below…