The six times in John's Gospel when Jesus says "I Am" followed by nothing else, which are four times when preaching (8:24,28,58; 13:19) plus two times when responding to the party that had come to apprehend Him (18:5,8), the words ἐγὼ εἰμί (Ego Eimi) translate אהיה (Ehyeh) of Ex 3:14.
Jesus' listeners might have understood that the expression in itself was a claim to divinity if Jesus said it in Hebrew in the middle of a dialogue carried out in Aramaic. Moreover, I hypothesize that Jesus did exactly that in his trial, as the High Priest and the scribes would certainly have understood the reference to Ex 3:14:
But He kept silent and did not answer. Again the high priest was
questioning Him, and saying to Him, "Are You the Christ, the Son of
the Blessed One?" And Jesus said, "I Am. And you will see the Son of
Man sitting at the right hand of Power and coming with the clouds of
heaven." (Mk 14:61-62).
But just as the composite scriptural quote following "I Am" in this passage was enough in itself to be understood as a claim to divinity, so was the text preceding "I Am" in Jn 8:58: "before Abraham came to be (genesthai)", as it implied Jesus' pre-existence since time immemorial before being conceived in human form.
The case that John has Ex 3:14 in mind when recording Jesus stating "I Am" followed by nothing else is further supported by understanding the falling to the ground of the soldiers that had come to arrest Jesus in response to his saying "I Am" in light of the rite of the feast of Atonement. For that, we must note two facts:
The term "fell" (epesan) is used 5 times by John in Revelation in the sense of "fall on their face" to worship: 5:8, 5:14, 7:11, 11:16 and 19:4.
By the time of Jesus, the proper Name of God in the third person revealed in Ex 3:15: YHWH, "He causes to be" if vocalized YaHWeH, was uttered by only one person, the High Priest, on only one day of the year, the feast of Atonement (Yom Kippur), 6 times when making a sacrifice for his own sins, one time when drawing the lot for the he-goats, and 3 times when loading the iniquities and transgressions of the sons of Israel on the he-goat to be sent to the desert (Lev 16:20-22). The prayer used by the High Priest for the latter function, and the people's response, are in the Mishna, tractate Yoma, chapter 6:
He then came to the he-goat which was to be sent away to
Azazeil and forcefully leans his hands on it and confesses. And so he
would say: Please O YHWH, they have done wrong they have transgressed
they have sinned before You - Your nation the House of Israel, Please,
O YHWH, forgive them for their doing wrong, for their transgressions
and for their sins, as is written in the Torah of Moshe Your servant:
“For on this day He will effect atonement for you to purify you before
YHWH” (Leviticus 16:30). And when the priests and the people who were
standing in the courtyard heard the fully pronunced Name come from the
mouth of the High Priest they would kneel, prostate themselves, fall
on their faces, and call out: Blessed be the Name of His glorious
kingdom for ever and ever. He gave it over to the one who was to lead
it [to Azazeil].
(The original response was Psalm 113:2 "Blessed be the Name of the LORD for ever and ever". It was later modified by the rabbis by changing "the LORD" to "His glorious kindgdom" and recorded that way in the Mishna).
From these data, the meaning of the fall to the ground of the party that had come to apprehend Jesus when He said "I Am" for the first time is crystal clear: Jesus is the High Priest who is carrying out the true Atonement prefigured by the rite in the Mosaic Law, and that at the time of loading the iniquities and transgressions of men on the victim that will carry them, pronunces the proper Name of God, with the difference, with respect to an ordinary High Priest, that:
since Jesus Himself is the victim, He bears and carries our iniquities and transgressions Himself,
since Jesus Himself is God, He pronounces the proper Name of God in the first person.
Finally, the third time when Jesus pronounces the proper Name of God in the first person as true High Priest of the true Atonement is not recorded in John's Gospel but in Mark's, in the reply to the High Priest quoted above. To validate the interpretation of this "I Am" as the third utterance of the divine Name in the first person by Jesus as High Priest of the true Atonement, we must note that, in the Jewish rite, immediately after the High Priest finished his prayer uttering the divine Name by a third time, the goat was taken to the desert. Similarly, immediately after pronouncing the third "I Am" in Mk 14:62, Jesus started to be spit, striken, mocked and slapped by the Jews (Mk 14:65).
Acknowledgment: I learned of this theological meaning of the passage from a site on the revealed Name of God by a Jewish scholar:
From the same site, I quote a passage in support of understanding אהיה (Ehyeh) of Ex 3:14 as the name of God in the first person, i.e. when said by God Himself.
Finally to the contributions of two 12th century exegetes who were
also Hebrew grammarians: Abraham Ibn Ezra and Rashbam. First to
Abraham Ibn Ezra, who often consulted his good friend Judah Halevi on
matters of exegesis and grammar. These exegetical discussions
evidently included the interpretation of Exodus 3:14, because not only
do they both correctly identify ehyeh as the Divine name in this
verse, but they both also find the meaning of this name in ehyeh asher
ehyeh and identify Yah as a related name. However, Ibn Ezra was bolder
than Halevi in certain very important respects and most especially in
regard to his grammatical analysis of the verse.
Briefly stated, Ibn Ezra identified the first ehyeh of ehyeh asher
ehyeh as a Divine name and proposed that the asher ehyeh fragment of
this declaration explains the meaning of the first ehyeh. He further
stated that ehyeh and YHWH are both proper names of God and even that
they have the same meaning, the only difference between them being
that ehyeh is in the first person while YHWH is in the third. Ibn
Ezra thereby implicitly identified ehyeh as the name YHWH when
employed by God in naming Himself. He did not separately mention the
ehyeh of 3:14b, but the similarity of his interpretation to Halevi's
strongly suggests that he did regard the ehyeh of 3:14b as the Divine
name and so presumably identical in meaning to the first ehyeh of
ehyeh asher ehyeh.
The second of the two grammarians, Rashbam, went one step further
again, in that he did specifically identify the ehyeh of 3:14b as the
first person form of the third person name YHWH, and so he explicitly
identified it as the name YHWH when used by God in naming Himself,
 Strickman N. and Silver A. (trans), Ibn Ezra's Commentary on the
Pentateuch: Exodus (Shemot), (NY: Menorah, 1997), p.64. In a footnote
to Ibn Ezra's comment on Ex.3:14, the authors explain his
interpretation as follows: "According to I.E., That I Am explains I
Am. In other words, God's name is not I Am That I Am. His name is I
Am, the meaning of which is, That I Am". Ibn Ezra's comment on
Ex.3:15 describes the name YHWH in the following terms: "Another name
meaning the same as the first one. However, one name (EHYH) is in the
first person and this name (YHWH) is in the third person".
 Propp W., Exodus 1-18, A New Translation with Introduction and
Commentary, The Anchor Bible, (NY: Doubleday, 1998), p.225