Sorry, no.
The relationship between individual organisms and gene pools is that of a specimen towards its collective of other biologically compatible specimens - and potential rivals for same, of course.
Individuals have genetic
compositions of their own, and they contribute to collective gene pools along with many other individuals. Individuals may (and do) take part of gene pools, and they are in a sense "built" out of elements of gene pools, but they do not
have whole gene pools in themselves, because they are individuals. Gene pools are groups.
Allow me to illustrate with this picture (Source:
Mechanisms of Evolution | OpenStax: Concepts of Biology | Study Guides )
Each marble represents a specimen, an individual. Each shade or hue of color represents a whole specific genetic composition.
While this picture does not convey that very well, each individual / marble except perhaps for identical twins will have some measure of variation of color / genetic composition from every other marble / individual.
In this illustration, the interior of the bottle represents a certain gene pool before some sort of bottleneck event greatly diminished the population available for potential mating. Perhaps grave famine, or some deadly epidemic, or very intense predation by other lifeforms.
After that event, the surviving population is much diminished, and so is the gene pool. It is not just that there are less specimens, but also that the potential genetic diversity among that population is much lesser than in the previous gene pool; there are less colors and less hues present (the brighter red marbles did not make it to the glass).
Those red marbles represent a certain amount of genetic diversity that was lost due to the bottleneck event.