That Jesus predicted the fall of Jerusalem and the destruction of the temple seems to be a fact of history.
People saw Jesus as a prophet, a successor to John the Baptist. They compared him to Elijah and Jeremiah. You can see why they compared him to Elijah. People believed both had worked wonders, even raised the dead. Also, they both came from up north.
Why did they compare Jesus to Jeremiah? The most obvious comparison lies in their correctly predicting the destruction of the temple. Jeremiah predicted the Babylonian destruction of the temple and Jesus predicted the Roman destruction of the temple.
We have trouble pinning down the exact words Jesus used, because all the gospels may have been finally written after the prediction had come true in 70 CE. So the gospel’s reporting may be influenced by the events themselves. For instance, Mark 13 and parallels may rely on either the event itself or on other Christian prophecies, which must have proliferated in the decade of the 60s as the events began to unfold. Still, even if we don’t know that the historical Jesus actually said of the temple, “not one stone will be left on another”, it remains likely that Jesus made some such prophecy.
My candidate for the authentic words of Jesus would be Luke 17:37: “They asked him, ‘Where, Lord?’ He said to them, ‘Where the body is, there will the eagles also be gathered together.'” We miss this as a prophecy of the destruction of Jerusalem by Rome, because translations usually say “vultures” instead of “eagles”. The word can mean either. But it becomes very clear what Jesus is saying if we think of eagles. Just as an eagle usually tops off our American flags, so the Roman army marched to war with standards tipped by eagles. So we should think of the body as Jerusalem and the eagles gathered round it as the Roman army. Jesus was saying that when they saw this, they would know his words were being fulfilled.
He had just said that there would be two people sleeping in the same bed. One would be taken and the other left. There would be two people grinding grain. One would be taken and one would be left. This is what would happen when Roman army gathered around the holy city. So the people who apply this passage to the Rapture and the Left-Behind idea just don’t get it. It is about death, particularly the kind of random death that comes about as collateral damage in war.
That the passage is a kind of riddle/parable and that it has the double meaning of eagles or vultures makes it exactly the kind of saying I would expect from the historical Jesus.