Here is the basic case:
DA 10:4 On the twenty-fourth day of the first month, as I was standing on the bank of the great river, the Tigris, 5 I looked up and there before me was a man dressed in linen, with a belt of the finest gold around his waist. 6 His body was like chrysolite, his face like lightning, his eyes like flaming torches, his arms and legs like the gleam of burnished bronze, and his voice like the sound of a multitude.
DA 10:7 I, Daniel, was the only one who saw the vision; the men with me did not see it, but such terror overwhelmed them that they fled and hid themselves. 8 So I was left alone, gazing at this great vision; I had no strength left, my face turned deathly pale and I was helpless. 9 Then I heard him speaking, and as I listened to him, I fell into a deep sleep, my face to the ground.
DA 10:10 A hand touched me and set me trembling on my hands and knees. 11 He said, "Daniel, you who are highly esteemed, consider carefully the words I am about to speak to you, and stand up, for I have now been sent to you." And when he said this to me, I stood up trembling.
DA 10:12 Then he continued, "Do not be afraid, Daniel. Since the first day that you set your mind to gain understanding and to humble yourself before your God, your words were heard, and I have come in response to them. 13 But the prince of the Persian kingdom resisted me twenty-one days. Then Michael, one of the chief princes, came to help me, because I was detained there with the king of Persia. 14 Now I have come to explain to you what will happen to your people in the future, for the vision concerns a time yet to come."
DA 10:15 While he was saying this to me, I bowed with my face toward the ground and was speechless. 16 Then one who looked like a man touched my lips, and I opened my mouth and began to speak. I said to the one standing before me, "I am overcome with anguish because of the vision, my lord, and I am helpless. 17 How can I, your servant, talk with you, my lord? My strength is gone and I can hardly breathe."
DA 10:18 Again the one who looked like a man touched me and gave me strength. 19 "Do not be afraid, O man highly esteemed," he said. "Peace! Be strong now; be strong."
When he spoke to me, I was strengthened and said, "Speak, my lord, since you have given me strength."
DA 10:20 So he said, "Do you know why I have come to you? Soon I will return to fight against the prince of Persia, and when I go, the prince of Greece will come; 21 but first I will tell you what is written in the Book of Truth. (No one supports me against them except Michael, your prince. 1 And in the first year of Darius the Mede, I took my stand to support and protect him.)
Here we see that Gabriel describes what some have seen to be a spiritual battle between angelic and demonic powers (although others see it as an angelic power wrestling with the human prince of Persia to bring about a change of mind).
In any case, a few points here jump out. At the beginning of the passage Daniel sees a figure that has striking parallels to the vision of Jesus in Revelation 1. Since Gabriel says that Michael had helped him some see this appearance as an appearance of Michael, and therefore link it with the vision in Revelation 1.
Daniel: Dressed in linen, with a belt of the finest gold around his waist. 6 His body was like chrysolite, his face like lightning, his eyes like flaming torches, his arms and legs like the gleam of burnished bronze, and his voice like the sound of a multitude.
Revelation: And in the midst of the lampstands one like a son of man, clothed with a long robe and with a golden sash around his chest. The hairs of his head were white like wool, as white as snow. His eyes were like a flame of fire, his feet were like burnished bronze, refined in a furnace, and his voice was like the roar of many waters.
On the other hand 10:13 describes Michael as ONE of the chief princes which seems to go against the notion that it is Christ.
DA 12:1 "At that time Michael, the great prince who protects your people, will arise. There will be a time of distress such as has not happened from the beginning of nations until then. But at that time your people--everyone whose name is found written in the book--will be delivered. 2 Multitudes who sleep in the dust of the earth will awake: some to everlasting life, others to shame and everlasting contempt. 3 Those who are wise will shine like the brightness of the heavens, and those who lead many to righteousness, like the stars for ever and ever. 4 But you, Daniel, close up and seal the words of the scroll until the time of the end. Many will go here and there to increase knowledge."
There are some strong parallels between this passage and John 5 in which Jesus says He will speak and the dead who are in their graves will rise to everlasting life or shame and everlasting contempt. Here Michael is called the great prince rather than just one of the chief princes.
Joh 5:28 Do not marvel at this, for an hour is coming when all who are in the tombs will hear his voice
Joh 5:29 and come out, those who have done good to the resurrection of life, and those who have done evil to the resurrection of judgment.
In Jude we see Michael again mentioned:
JUDE 1:8 In the very same way, these dreamers pollute their own bodies, reject authority and slander celestial beings. 9 But even the archangel Michael, when he was disputing with the devil about the body of Moses, did not dare to bring a slanderous accusation against him, but said, "The Lord rebuke you!" 10 Yet these men speak abusively against whatever they do not understand; and what things they do understand by instinct, like unreasoning animals--these are the very things that destroy them.
Here Michael is seen disputing with Satan. Now some have pointed out that he says "the Lord rebuke you." In other words, why would he say that if he were Jesus? However, this is actually quoting from Zechariah originally
ZEC 3:1 Then he showed me Joshua the high priest standing before the angel of the LORD, and Satan standing at his right side to accuse him. 2 The LORD said to Satan, "The LORD rebuke you, Satan! The LORD, who has chosen Jerusalem, rebuke you! Is not this man a burning stick snatched from the fire?"
So here we see the LORD saying "the Lord rebuke you" So it doesn't exactly rule it out, and may even strengthen the association. Again in this context, as with the others, we see Michael involved in a spiritual struggle.
The reference to Moses body is believed to be a quote from a lost portion of an extra-biblical book, the assumption of Moses.
Moses and Elijah were both present at the transfiguration suggesting that Moses' body was resurrected, as there is no reference to Moses being out of body and Elijah was taken directly to heaven without dying.
Again we see Michael in Rev. 12:7
REV 12:7 And there was war in heaven. Michael and his angels fought against the dragon, and the dragon and his angels fought back. 8 But he was not strong enough, and they lost their place in heaven. 9 The great dragon was hurled down--that ancient serpent called the devil, or Satan, who leads the whole world astray. He was hurled to the earth, and his angels with him.
Here we see Michael as in charge of the angels, clearly again in conflict.
Thessalonians does not mention Michael, but does speak of the voice of an archangel.
1 Thess. 4:16 For the Lord himself will come down from heaven, with a loud command, with the voice of the archangel and with the trumpet call of God, and the dead in Christ will rise first. 17 After that, we who are still alive and are left will be caught up together with them in the clouds to meet the Lord in the air. And so we will be with the Lord forever. 18 Therefore encourage each other with these words.
Here we see that it says that the Lord will come down with a loud command and the voice of the archangel.
Some see the voice as that of Jesus, also spoken of in John 5 as before, and some see it as an angel accompanying Him.
As to the term for archangel, it seems it is up to those who take the position that it does not indicate that Jesus is an angel but merely over the angels to demonstrate that the prefix αρχ- is used in that way. In the example of ἀρχιερεύς, chief priest, the person was still a priest, though over the others. If the same model were followed Michael would still be an angel, though over the others. Is there linguistic evidence of the opposite? One who is over but does not partake in that nature? I don't know. I tend to think that the prefix is to distinguish superiority WITHIN a class more than simply to indicate status above a class. It would be much like our English usage "arch-enemy" etc. An arch-enemy is one's chief enemy. He is still an enemy.
Overall I think there is some evidence that could point to the connection, but the idea that Jesus is one of the chief priests does not fit, nor has it been demonstrated than an ἀρχαγγέλοs is not an angel. So with that in mind I would say it is not the case.
