Matthew's Gospel and Luke's Gospel, each with a different approach, agree that Jesus was born in Bethlehem. We have no other sources concerning the birth of Jesus. “If we abide by the sources, it is clear that Jesus was born in Bethlehem and grew up in Nazareth.” (1)
“Mary laid her newborn child in a manger (Cf. Lk 2:7). From this detail it has been correctly deduced that Jesus was born in a stable, in an inhospitable - one might even say unworthy - place, which nevertheless provided the necessary privacy for the sacred event. In the area around Bethlehem, rocky caves had been used as stables since ancient times.
As early as Justin Martyr (+165) and Origen (+ c.254), we find the tradition that Jesus was born in a cave, which Christians in Palestine could point to. The fact that after the expulsion of the Jews from the Holy Land in the second century, Rome turned the cave into a shrine of Tammuz-Adonis, thereby evidently intending to suppress the Christian memorial cult, confirms the age of this shrine and also shows how important it was thought to be by the Romans. Local traditions are frequently a more reliable source than written records. So a considerable measure of credibility may be assigned to the tradition that Bethlehem was Jesus’ birthplace, a tradition to which the Church of the Nativity also bears witness.” (2)
(1) J. Ratzinger, Benedict XVI, Jesus of Nazareth: the Infancy Narratives, Image Books, 2012, p. 65
(2) Ibidem p. 66-67