Advaita Vedanta is based on the Upanishads and the Bhagavad-Gita.
Famous Indologist and Buddhist, Patrick Olivelle says on the Upanishads,
The scholarly consensus, well founded I think, is that the
Brhadaranyaka and the Chandogya are the two earliest Upanisads. We
have seen, however, that they are edited texts, some of whose sources
are much older than others. The two texts as we have them are, in all
likelihood, pre-Buddhist; placing them in the seventh to sixth
centuries BCE may be reasonable, give or take a century or so. The
three other early prose Upanisads—Taittiriya, Aitareya, and
Kausitaki—come next; all are probably pre-Buddhist and can be assigned
to the sixth to fifth centuries BCE. The Kena is the oldest of the
verse Upanisads and contains many of the themes, such as the search
for the one god who is both the creator and the agent of liberation
for humans, that recur in the four subsequent verse Upanisads. Of
these, the oldest is probably the Katha, followed by Isa,
Svetasvatara, and Mundaka. All exhibit strong theistic tendencies and
are probably the earliest literary products of the theistic tradition,
whose later literature includes the Bhagavad Gita and the Puranas. All
these Upanisads were composed probably in the last few centuries BCE.
(The Early Upanishads, pages 12-13)
We shall address the Bhagavad-Gita now.
Prof Max Muller writes, : ”Professor Wilson, indeed, thought that
there was an allusion to Buddhism in the Gita. But his idea was based
on a confusion between the Buddhists and the Carvakas or materialists.
Failing that allusion, we have nothing very tangible but the
unsatisfactory 'negative argument' based on mere non-mention of
Buddhism in the Gita. Either the Gita and Buddhism were alike the
outward manifestation of one and the same spiritual upheaval which
shook to its centre the current religion, the Gita being the earlier
and less thorough-going form of it; or Buddhism having already begun
to tell on Brahmanism, the Gita was an attempt to bolster it up. I
do not accept the latter alternative, because I cannot see any
indication in the Gita of an attempt to compromise with a powerful
attack on the old Hindu system; while the fact that, though strictly
orthodox, the author of the Gita still undermines the authority, as
unwisely venerated, of the Vedic revelation; and the further fact,
that in doing this, he is doing what others also had done before him
or about his time; go, in my opinion, a considerable way towards
fortifying the results of the negative argument already set forth.
To me Buddhism is perfectly intelligible as one outcome of that play of thought on high spiritual topics, which in its other, and as we may
say, less thorough-going manifestations, we see in the Upanishads and
the Gita. The Upanishads, with the Gita, and the precepts of Buddha
appear to me to be the successive embodiments of the spiritual thought
of the age, as it became more and more dissatisfied with the system
of mere ceremonial then dominant.” (The Sacred Books of the East, Vol
8, Pages 43–44)
Prof Muller concludes,
To my mind having approached Buddhism after a study of the ancient
religion of India, the religion of the Veda, Buddhism has always
seemed to be, not a new religion, but a natural development of the
Indian mind in its various manifestations”(Chips from a German
Workshop, i, p. 434).
After proving Gita and the Upanishads are Pre-Buddhist, we shall now proceed to specific terms in Advaita which were inferred from Buddhism. I have written an entire answer on the terms used here 1. To conclude in short,
It is evident from the above explanation that Gaudapada's philosophy
is. It Buddhist, but purely Upanishadic 'why, then, it may be
asked, should he have adopted Buddhistic expression at all? The
answer is that the exigencies of his time must have made him use
Bauddha terminology, even as the Hindu monks who preach Vedanta in
the countries of the west today feel the necessity of clothing their
thoughts in Christian expressions.(Ashokan N, Mahayana Buddhism And
Early Advaita Vedanta, Chapter 5.10)
