Two things worth bringing up. First -- Caesar did receive and accept the military honour of the corona civica, often translated as the "civic crown," but this was a separate issue. The civic crown was a highly respected military award which does not constitute any monarchic claim.

The story of Caesar being offered a crown is recorded by Plutarch, who explains (in chapter 61 of his life of Caesar) that Caesar was offered a crown several times by Marcus Antonius (Shakespeare's Antony), who was one of the runners at a festival called the lupercalia. Caesar refused the crown when it was offered. Plutarch appears to indicate that Caesar was hoping that the crowd would want him to take it, but this is a problematic take on the situation. Plutarch goes further to say that some of Caesar's statues were decorated with crowns, but tribunes removed them, and Caesar grew angry and punished them.

Note that the "crown" at hand is perhaps better translated as diadem, which would be a ribbon tied around the head rather than the metal crown we think of from later European history. Plutarch tells us that it was tied with a laurel branch. The diadem was a very old Near-Eastern symbol of monarchy, which assimilated with the Greek tradition of wearing wreaths, and became a Hellenistic symbol of divine kingship.

It's in the sources.

Suetonius

and at the Lupercalia, when the consul Antony several times attempted to place a crown upon his head as he spoke from the rostra, he put it aside and at last sent it to the Capitol, to be offered to Jupiter Optimus Maximus.

Plutarch

Accordingly, after he had dashed into the forum and the crowd had made way for him, he carried a diadem, round which a wreath of laurel was tied, and held it out to Caesar. Then there was applause, not loud, but slight and preconcerted. But when Caesar pushed away the diadem, all the people applauded; and when Antony offered it again, few, and when Caesar declined it again, all, applauded.

Appian

Nor were the attempts to claim royal honours for him brought to an end even thus, for while he was in the forum looking at the games of the Lupercal, seated on his golden chair before the rostra, Antony, his colleague in the consulship, who was running naked and anointed, as was the priests' custom at that festival, sprang upon the rostra and put a diadem on his head.

I'm sure there are others. I suppose you could ask whether the earliest sources (I think Appian here) made that scene up and everyone else just copied him, but you run out of history real quick if you start thinking that way.

I think the question in academic circles isn't whether Antony offered Caesar the crown, but whether it was a plot, and was Caesar in on it - I mean was it Antony trying to get Caesar to claim the title "King" and give grounds for his assassination, or was it Caesar trying out the title and, based on the crowd's reaction, deciding it was a bad idea, or was it Antony being an idiot? I don't know anyone that says this incident didn't happen.