The rescue operation in Berchtesgaden, Bavaria, continued into its fourth day as two doctors tried to reach the wounded 52-year-old.
Westhauser, who works at the Karlsruhe Institute of Technology, was injured nearly 1,000 metres below ground in a rock fall in the Riesending caves in the Bavarian Alps on Sunday.
On Monday, emergency services managed to make contact with him, but police said the man was not able to be moved and a specialist rescue team was called in from Switzerland.
And on Wednesday afternoon, one doctor and three rescue team members descended to the halfway point and were resting on the third of five bivouacs set up along the route, Stefan Schneider, deputy head of the Bavarian mountain rescue said.
Another group with a second doctor had begun the descent and hoped to reach the third bivouac before a rest was needed.
The long vertical descent is arduous, and the Bavarian mountain rescue team cited the necessity of planning breaks throughout the journey to give the doctors enough time to recover.
"I find it absolutely exceptional that someone should have to go down there," said Schneider.
He called the effort undertaken by the doctors a "milestone" in the history of rescue work.
Currently, 20 people are in the cave and around four kilometres of safety rope has been laid out.
A mountain rescue team arrived from northern Italy on Tuesday and worked in the cave throughout the night to optimize the route and reinforce safeguards in preparation for the doctors' journey.
On Wednesday afternoon Westhauser's condition remained stable, but he was said to be suffering from a head injury.
Once the doctors have reached him and completed their examination, it will be determined how to best handle the ascent with the patient.
The process of transporting the explorer is expected to take six days in optimal conditions and will present a variety of challenges. Certain parts of the cave bottleneck only allow one person at a time to pass.
It is possible that the patient may be lifted with the help of pulleys.