There are two reasons from the Old Testament that Bethlehem would be significant as the birthplace of Jesus:-
First Samuel tells us that Bethelehem was the home town of the future King David:
1 Sam 17:12,15: Now David was the son of that Ephrathite of Bethlehemjudah, whose name was Jesse; and he had eight sons: and the man went among men for an old man in the days of Saul ... But David went and returned from Saul to feed his father's sheep at Bethlehem.
1 Sam 20:6: If thy father at all miss me, then say, David earnestly asked leave of me that he might run to Bethlehem his city: for there is a yearly sacrifice there for all the family.
Then Micah 5:2 tells us that a future king would also come from Bethlehem:
Micah 5:2: But thou, Bethlehem Ephratah, though thou be little among the thousands of Judah, yet out of thee shall he come forth unto me that is to be ruler in Israel; whose goings forth have been from of old, from everlasting.
Adrian Hastings says in A World History of Christianity, page 15, that some Jews looked forward to a messianic figure as a political leader of the line of David. It seems that if the Jews were to be convinced that Jesus was that Messiah, he would have to be descended from David and born in Bethlehem.
Mark, the first New Testament gospel to be written, makes no mention of Bethlehem and simply refers to Jesus of Nazareth. Early Christian converts who relied on Mark's Gospel had no reason to believe that Jesus was born in Bethlehem, and would likely have believed that Nazareth was his birthplace.
Although its author would have known that Luke's Gospel placed the birth of Jesus in Bethlehem, John appears to suggest that this was not the case.
John 7:41-42: Others said, This is the Christ. But some said, Shall Christ come out of Galilee? Hath not the scripture said, That Christ cometh of the seed of David, and out of the town of Bethlehem, where David was?
John 7:52: They answered and said unto him, Art thou also of Galilee? Search, and look: for out of Galilee ariseth no prophet.
So, we have one gospel whose author was unaware that Jesus was born in Bethlehem and one gospel whose author seems not to have believed that Jesus was born in Bethlehem.
We then have two gospels, Matthew and Luke, that place the birth of Jesus in Bethlehem, although the two nativity stories could hardly be more different. Uta Ranke-Heinemann says in Putting Away Childish Things, page 7, the nativity accounts in the Gospels of Matthew and Luke are, with respect to time, place, and circumstances, a collection of legends, a view shared by many other New Testament scholars. This is to say that the two authors, writing independently of each other, knew the importance of Jesus being born in Bethlehem and wrote to that effect. Matthew 2:6 is based on the words of Micah 5:2.