John 6:4 is usually held to be something like
"Now the Jewish Feast of the Passover was near." (Berean Study Bible)
The reason to be skeptical of John 6:4 is it seems to 1. not fit with the context of John as well as linguistically being odd, as Revelation Lad notes in his answer.
"If it were the Passover, the Jews and Jesus should be in Jerusalem,
not Capernaum. If it were the Passover, it would be before the barley
harvest, and barley bread would not be available (cf. John 6:9). If it
were the Passover, the Jews would not be eating leavened bread, like
barley bread. If it were the Passover, it should not be called the
Passover of the Jews as it was the Passover of the LORD."
And 2., it seems to make John not fit with the synoptic Gospels.
This motivates a skeptical eye towards the text. We have 3 options if there is indeed something wrong here, and we want to see if 6:4 can fit better with both the immediate context and the synoptics.
Either a) one's understanding of what's happening in the text is wrong, or b) there's a corruption, or, c) if one doesn't hold to inerrancy, the text is just wrong.
So if one does hold to inerrancy, either there's a misunderstanding of the line itself, or manuscript corruption.
If we look for historical evidence that this was not understood to be the Passover, we find it with the early Church Fathers, some specifics of which are found in John 6:4 - He Walks With Us.
It cites Clement of Alexandria's Stromata, I, 21, 145.
"And Jesus was coming to His baptism, being about thirty years old,”
and so on. And that it was necessary for Him to preach only a
year, this also is written: “He hath sent Me to proclaim the
acceptable year of the Lord.” This both the prophet spake, and the
It also cites Irenaeus of Lyon's lack of quoting John 6:4 when making an argument for 3 Passovers, when it's an obviously relevant verse to quote.
It cites Origen's First Principles (as well as Homilies on Luke and Commentary on John).
"[Jesus] taught only during a year and some months"
as well as Tyconius.
Furthermore, there are some manuscripts that omit the verse, apparently 472 (perhaps with a few others) and 1634 (see Should John 6:4 Be Considered a Spurious Insertion into John’s Gospel?).
"Next comes the number of the manuscript in which the verse is omitted
(472), with “pc” following, which stands for Latin pauci, meaning “a
few,” and indicating that there are a few other very minor manuscripts
which lack the verse but none earlier than the one listed and
significant enough to list. [...] in the final update of NA27, the
apparatus has #1634 for the minuscule that lacks the verse, and this
is likewise what is found in the electronic editions of the NA27
(e.g., in Accordance Bible Software and Logos Bible Software). More
than likely, the final update of NA27 reflects the editors’ opinion
that minuscule 1634 was a better representation of the few, late
manuscripts that lacked the verse."
From the REV commentary here:
"There is also some manuscript and evidence from the Church Fathers
that “Passover” was added. There are a few manuscripts that omit the
whole verse of John 6:4 (see Nestle and Aland; Novum Testamentum
Graece, 26th edition, 1979, apparatus on John 6:4). That is entirely
possible because the verse does not fit in the context. However, there
is also evidence that the word “Passover” was not in the early
manuscripts, which would mean that the original text read “Now the
feast of the Jews was near.” Statements made by Origen, Cyril of
Alexandria, and Irenaeus are consistent with the fact that they had
manuscripts that did not have the Passover in John 6:4 (W. Cummins,
The Acceptable Year of the Lord, Scripture Consulting, 2005, p. 66).
If John 6:4 is just saying that “a feast of the Jews was near,” there
were a number of smaller feasts in the Jewish calendar, and the Law
did not require Jews to go to Jerusalem during those feasts.
There are a couple of different ways that the word “Passover” could
have been added to the text. One is quite accidental. If the original
text just said, “a feast of the Jews was near,” it would be entirely
possible that a scribe who thought that feast was Passover wrote
“Passover” in the margin of his manuscript. Scribes wrote notes in the
margin of their manuscripts just as Christians today write in the
margins of their Bibles. If that happened, then another scribe could
have thought “Passover” was left out of the last copy, and so copied
it into his next manuscript.
Another reason, less likely and less innocent, that John 6:4, or the
word “Passover” might have been added was in the early centuries after
Christ there were debates about the length of Jesus’ ministry. Anyone
who studies the manuscripts knows that there were a number of topics
that were hotly debated by the early Church, and it was not uncommon
for one side to alter a manuscript to support their argument (cp. Bart
Ehrman, The Orthodox Corruption of Scripture). Thankfully, today,
especially with the help of computers that help textual critics to
compare the over 5700 Greek manuscripts, it is usually easy to spot
those intentional changes and expunge them from the modern Greek text,
but a few manuscript discrepancies are still the subject of debate.
The length of Jesus’ ministry was a topic of debate in the centuries
after Christ, and John 6:4 could be one of the verses that was altered
to support a position."
There are 2 problems with the manuscript evidence, though. First, one has to hold the earlier manuscripts which do have 6:4 were done after the change, and that puts the change very early on, and that the ones without 6:4 are descended from manuscripts that didn't have this corruption. To be compelling, then, this would require a further argument about the family of manuscripts those are descended from, and why we are to think they are significant.
Also, the evidence cited by the early Church Fathers could fit both different manuscripts or a different understanding of the line by the early commentators.
To decide which of these is more plausible, you need to have some alternate interpretation of John 6:4 which keeps the wording, such as Revelation Lad outlines in his answer (tl ; dr the Passover being referenced is the Pesach Sheni, which was for those who could not celebrate the normal Passover).
Conclusion: yes, it's possible either the word 'Passover' or the entire line was added at some point, and there is some evidence to support it. As the other answers point out, however, this does not mean it is conclusive, nor even probable, and the strongest evidence for this view ('Passover' doesn't seem to fit internally or with the synoptics, and some writings from early Church Fathers) also supports the idea that the Passover being referenced was the Pesach Sheni.