You can find everything beautifully explained in this website on brass instruments: https://newt.phys.unsw.edu.au/jw/brassacoustics.html
A short version is that for ultrasonic (>20 kHz) vibrations you would need to increase the air pressure made by the mouth. That can be done by either putting more air through the mouth or making the volume through bigger. Alternatively, you could think of the "embouchure" being very tiny to make the pressure very high (I will do some quick math here and assume a linear relationship between the pressure and with a typical value of 1kPA to make a trumpet sound around 1kHz, maybe 20 times the pressure will lead to ultrasonic soundwaves).
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8313787/ Here you can see that the damage threshold of the tympani is around 30-100 kPa, so I would speculate that you would need to apply such an absurd amount of pressure to whistle ultrasonically that you would rupture your timpani at the same time.
I am neither a clinical expert or acoustics expert, so those numbers are just from "fun" calculations based on the linked evidence, but I hope there are more publications about what is the true range that a person can actually achieve "humanly".
EDIT: It has come to my attention from the comments that I did not clarify precisely what I meant with my analogy to brass instruments.
https://en.wikipedia.org/wiki/Physics_of_whistle
The wikipedia article is deemed to deep for it being a wikipedia article, but it has some interesting aspects of how whistling occurs in humans:
First, it is defined as a fluid dynamics problem:
The physical theory of the sound-making process is an example of the application of fluid dynamics or hydrodynamics and aerodynamics.
Regarding human whistling, the wikipedia article is confusing but a study from Wilson et. Al. https://pubs.aip.org/asa/jasa/article-abstract/50/1B/366/745861/Experiments-on-the-Fluid-Mechanics-of-Whistling?redirectedFrom=fulltext points to a similar behavior when using a model based on a cylinder with different holes (for input and output).
While it is noted that the fact that I used the trumpet/brass as an example being a farfetched example, as I find further examples where the explanation is done more through fluid dynamics I think it is more difficult for me to make an informed assessment, but I still think that the amount of airflow to have supersonic sound out of a person is unlikely.