There are some clues in these verses [Daniel 10:2-4] that may make it possible for us to estimate rather precisely when this appearance of God came to Daniel. He says that he had been mourning and fasting for three “full” weeks and that then God appeared to him on the twenty-fourth day of the first month–Nisan (10:4). Given the close proximity of these statements, the implication is that the twenty-fourth day of the first month took place immediately at the end of the three weeks of fasting. The original language uses an idiom here to indicate that the weeks were “full.” Full weeks come to an end after seven days; they end on Sabbath, the seventh day. Since this vision appeared to Daniel at the end of three full weeks, it must also have come to him on a Sabbath day. That means that this final prophecy of the book of Daniel was most likely given on a Sabbath. This is the only vision in the book that we can date with such precision.
Shea, William H., Daniel, 2 vols., Boise, Idaho: Pacific Press Publishing Association, 1996, vol. 2, p. 172.