when the Child was born in Bethlehem, since Joseph could not find a lodging in that village, he took up his quarters in a certain cave near the village; and while they were there Mary brought forth the Christ and placed Him in a manger, and here the Magi who came from Arabia found Him.Justin Martyr. Dialogue with Trypho, Chapter 78
With respect to the birth of Jesus in Bethlehem, if any one desires, after the prophecy of Micah and after the history recorded in the Gospels by the disciples of Jesus, to have additional evidence from other sources, let him know that, in conformity with the narrative in the Gospel regarding His birth, there is shown at Bethlehem the cave where He was born, and the manger in the cave where He was wrapped in swaddling-clothes. And this sight is greatly talked of in surrounding places, even among the enemies of the faith, it being said that in this cave was born that Jesus who is worshipped and reverenced by the Christians.
(Both finds thanks to Wikipedia)
I would consider Matthew a more reliable witness he being on hand and all rather than someone a century and more later.
As for the writings of the rabbis....one has to sift an awful lot of sand to find a flecks of gold.