The answer will depend on the Vedanta school that is being followed.
The position of Advaita Vedanta school is given below.
Bodies appear and disappear, but not so the Atman which ever is. ..
Commentary by Swami Chidvabananda on Gita 2.12
This suggests that the Atman present in all is eternal. As far as I can see the Advaita school does not distinguish between the individual essence of a jiva and the universal aspect. Atman is present universally and is eternal.
The Vishistadvaita school arrives at a different conclusion. This school concludes that difference between the Lord and the Jivas is real and the Jivas are eternal because the universe and the Jivas form the body of the Lord. Thus one can say that the principle of God is common between Jivas and the Lord and it is this universal aspect that is eternal.
Indeed I, the Lord of all, who is eternal, was never non-existent, but
existed always. It is not that these selves like you, who are subject
to My Lordship, did not exist; you have always existed. It is not that
'all of us', I and you, shall cease to be 'in future', i.e., beyond
the present time; we shall always exist. Even as no doubt can be
entertained that I, the Supreme Self and Lord of all, am eternal,
likewise, you (Arjuna and all others) who are embodied selves, also
should be considered eternal.
The foregoing implies that the difference between the Lord, the
sovereign over all, and the individual selves themselves, are real.
This has been declared by the Lord Himself. For, different terms like
'I', 'you'. 'these', 'all' and 'we' have been used by the Lord while
explaining the truth of eternality in order to remove the
misunderstanding of Arjuna who is deluded by ignorance.
Commentary on Gita 2.12 by Sri Ramanuja translated by Swami Adidevananda