Due to the large number of known ziggurats, the exact location of the original Tower of Babel has been debated.

The precise site of the ancient tower of Babel is a matter of uncertainty, for there are possibilities among the remnants of several ruins in the region.
Many writers, following Jewish and Arab traditions, locate the tower ruins at Borsippa (the “Tongue Tower”), about eleven miles southwest of the northern portion of Babylon (formerly a suburb of the city).
Others identify the site with Etemen-an-ki (“the temple of the foundation of heaven and earth”), which is located in the southern sector of the city near the right bank of the Euphrates river.1)Wayne Jackson, “The Tower of Babel: Legend or History,” https://www.christiancourier.com/articles/140-the-tower-of-babel-legend-or-history

At the ancient site of Borsippa (meaning “Tongue Tower”) today stands the largest Ziggurat that remains in existence. Many have assumed this to be the original Tower of Babel. One archaeologist wrote:

I am pleased to see that Dr. Kraeling maintains the identification of the Tower of Babel with Birs Nimrud. That has been my view ever since I first saw the remarkable ruins of Birs Nimrud in 1889. They are more striking to the eye than anything in Babylon, and they lie sufficiently near to Babylon to make the ordinary man connect them with the famous name of Babel, for indeed Borsippa must have seemed to him no more than a suburb of the great city.2)John P. Peters, “The Tower of Babel at Borsippa,” Journal of the American Oriental Society, Vol. 41 (1921), p. 157; accessible at http://www.jstor.org/stable/593716

However, there was clearly a distinction between Borsippa and Babylon, as is indicated in Nebuchadnezzar’s Foundation Stone Inscription, which states:

Highly have I exalted their cities;
(but) above Babylon and Borsippa
I have not added a city
in the realm of Babylonia
as a city of my lofty foundation.3)Text of Nebuchadnezzar’s Foundation Stone Inscription (Column 7:31-35) accessible at http://mcadams.posc.mu.edu/txt/ah/Assyria/Inscrb00.html

Grant Jeffery held the impression that Borsippa was the site of the Tower of Babel, which he expressed his evidence by citing an inscription from the site.

The tower, the eternal house, which I founded and built.
I have completed its magnificence with silver, gold, other metals, stone, enameled bricks, fir and pine.
The first which is the house of the earth’s base,
The most ancient monument of Babylon; I built and finished it.
I have highly exalted its head with bricks covered with copper.
We say for the other, that is, this edifice, the house of the seven lights of the earth,
The most ancient monument of Borsippa [Chaldean meaning tongue-tower].
A former king built it, (they reckon 42 ages) but he did not complete its head.
Since a remote time, people had abandoned it, without order expressing their words.
Since that time the earthquake and the thunder had dispersed the sun-dried clay.
The bricks of the casing had been split, and the earth of the interior had been scattered in heaps. Merodach, the great god, excited my mind to repair this building. I did not change the site nor did I take away the foundation.
In a fortunate month, in an auspicious day,
I undertook to build porticoes around the crude brick masses, and the casing of burnt bricks.
I adapted the circuit, I put the inscription of my name in the Kitir of the portico.
I set my hand to finish it. And to exalt its head.
As it had been in ancient days, so I exalted its summit.4)Nebuchadnezzar’s Inscription trans. Professor Andrew Oppert; as cited by Grant R. Jeffrey, The signature of God, p. 40-41

This inscription refers to the “The most ancient monument of Babylon” as well as “The most ancient monument of Borsippa.” Whether this is to be interpreted as the same temple or separate temples is difficult to determine. However, of the temple in Borsippa it is said, “A former king built it…) but he did not complete its head.” As was commented before, Scripture does indicate that the Tower of Babel was completed, but not the city. Also we will see that the Bebylonian ziggurat was built completely a number of times.
The Babylonian Talmud also identifies a distinction between Borsippa and Babylon.

It has been taught. R. Nathan said: They were all bent on idolatry. [For] here it is written, let us make us a name; whilst elsewhere it is written, and make no mention of the name of other gods: just as there idolatry is meant, so here too. R. Jonathan said: A third of the tower was burnt, a third sunk [into the earth], and a third is still standing. Rab said: The atmosphere of the tower causes forgetfulness. R. Joseph said: Babylon and Borsif are evil omens for the Torah. What is the meaning of Borsif? — R. Assi said: An empty [shafi] pit [bor].5)Babylonian Talmud, tractate Sanhedrin 109a accessed at http://come-and-hear.com/sanhedrin/sanhedrin_109.html

Here, the name Borsippa is wrote “Borsif” phonetically from Akkadian barsip, where in Hebrew the letter פ makes the “ph” sound and can be transliterated as “p,” “ph,” or “f.” This expression of connecting Borsippa “the Tongue Tower” with what the Jews defined as “an empty pit,” seems to be a description of the ziggurat that was in Babylon. This ziggurat was no longer in existence at the time the Talmud was written beyond the evidence of a great hole in the ground where it once stood, which is what the Talmud describes as “an empty pit.” This ziggurat was built, destroyed and rebuilt multiple times. “The ziggurat at Babylon was demolished by Xerxes in 472 B.C., and though Alexander cleared the rubble prior to its restoration this was thwarted by his death. The bricks were subsequently removed by the local inhabitants, and today the site of Etemenanki is a pit (Es-Sahn) as deep as the original was high.”6) D.R.W. Wood, New Bible Dictionary (Downers Grove, IL: Intervarsity Press, c1962, c 1982, 1996), S. 109
It was reported that “An inscribed foundation stone has been recovered, which details Nebuchadnezzar’s plan to have the Borsippa ziggurat built on the same design as that at Babylon, of which only the foundation survives.”7)http://www.cemml.colostate.edu/cultural/09476/iraq05-029.html Thus the Babylonian ziggurat is the older one which Borsippa was built as a duplicate of. As we saw, all the historical documents identified the Tower of Babel with Babylon.