Mark 7 contains a very curious story:
24 From there he set out and went away to the region of Tyre.[g] He entered a house and did not want anyone to know he was there. Yet he could not escape notice, 25 but a woman whose little daughter had an unclean spirit immediately heard about him, and she came and bowed down at his feet. 26 Now the woman was a Gentile, of Syrophoenician origin. She begged him to cast the demon out of her daughter. 27 He said to her, “Let the children be fed first, for it is not fair to take the children’s food and throw it to the dogs.” 28 But she answered him, “Sir,[h] even the dogs under the table eat the children’s crumbs.” 29 Then he said to her, “For saying that, you may go—the demon has left your daughter.” 30 So she went home, found the child lying on the bed, and the demon gone.Mark 7:24-30
This story is repeated in Matthew 15:
21 Jesus left that place and went away to the district of Tyre and Sidon. 22 Just then a Canaanite woman from that region came out and started shouting, “Have mercy on me, Lord, Son of David; my daughter is tormented by a demon.” 23 But he did not answer her at all. And his disciples came and urged him, saying, “Send her away, for she keeps shouting after us.” 24 He answered, “I was sent only to the lost sheep of the house of Israel.” 25 But she came and knelt before him, saying, “Lord, help me.” 26 He answered, “It is not fair to take the children’s food and throw it to the dogs.” 27 She said, “Yes, Lord, yet even the dogs eat the crumbs that fall from their masters’ table.” 28 Then Jesus answered her, “Woman, great is your faith! Let it be done for you as you wish.” And her daughter was healed instantly.Matthew 15:21-28
The gist of the translation is very clear; Jesus makes an analogy that helping her would be like taking food from his children and giving it to dogs. No matter how you slice it, this is an insult, and the woman's response would be considered groveling. This is the impression that some people have of the Jews, and it is very anti-semitic.
The problem is, it likely never happened. Jesus – if he was indeed from God, or even Jewish – would be well aware of the central mission of Israel as a Light to the Nations. Therefore, for him to insinuate that the gentile nations were somehow unworthy of his guidance violates God's will as stated in the Torah.
Before the issue is raised, the standard Christian explanation is that Jews think of gentiles as Dogs:
Non-Jews were considered so unspiritual that even being in their presence could make a person ceremonially unclean (John 18:28).
(Cross reference with John 18:)
28 Then they took Jesus from Caiaphas to Pilate’s headquarters. It was early in the morning. They themselves did not enter the headquarters, so as to avoid ritual defilement and to be able to eat the Passover.John 18:28
However, when considering Israel's role as a Light to the Nations this answer becomes unsatisfactory.
Upon raising the issue of A Light to the Nations it becomes important to rewrite this story so that Jesus is not seen as insulting to the woman. The Christian then reverses his claim that Jesus was justified in his approach by flipping the story on it's head and claiming that comparing the woman to a dog was a term of endearment:
The exact word Jesus used here, in Greek, was kunarion, meaning “small dog” or “pet dog.” This is a completely different word from the term kuon, used to refer to unspiritual people or to an “unclean” animal.https://gotquestions.org/Canaanite-woman-dog.html
The only issue with this is that kunarion still means dog.
Let’s interject ourselves into the story. How would you feel if someone implied that you and your little girl were worthless pagan dogs? However, notice the remarkable response of this amazingly humble woman. She agreed with Jesus’ assessment without defensiveness while continuing her plea.http://www.scborromeo.org/papers/The%20Canaanite%20Woman.pdf
Most commentaries will agree, comparing the woman to a dog, even a supposedly cute puppy-dog, really wasn't a nice thing to say, and was totally out of line with who Jesus should have known the Jewish people to be.