Some scholars think it was Sept 3-5BC although there is no proof, they think the "star of Bethlehem was a comet which passed very close to earth around the time Jesus was born, this is backed up by Chinese astrology charts of trhe same era
The date December 25th was chosen because it was already a pagan festival and the christain leaders decided people were more likely to celebrate it if it was already a festival.
If they had chosen another date then people may have ignored it.