Were the wise men at the manger when Jesus was born?

According to the Bible the wise men were not at the manger when Jesus was born.  The details of Jesus’ birth are found in Luke 2 and the wise men are nowhere near it.  Eight days after Jesus was born, He was not in Egypt but at Jerusalem being circumcised (Luke 2:21).  The wise men did not enter until about 2 years later.  In Matthew 2 we see that Jesus was called a young child, they were in a house in Bethlehem, and Herod killed the children 2 years and younger based on what the wise men told him.  So according to the Bible Jesus was about 2 when they came to give Him their presents.