Is the city of David where Jesus was born?

His family was from Bethlehem where he was also born. He was the seventh son of Jesse of Bethlehem

The Lord said unto Samuel, How long wilt thou mourn for Saul, seeing I have rejected him from reigning over Israel? Fill your horn with oil, and go, I send you to Jesse of Bethlehem, because of his children I have seen a king.

1 Samuel 16: 1