What Genesis actually says
The Nephilim appear in Genesis shortly before the flood. The passage links them to the difficult account of the sons of God and the daughters of men, and calls them mighty people from ancient times (Genesis 6:1-4). But it does not explain exactly who they were, what happened to them after death, or whether they became evil spirits.
Christians have long disagreed about the sons of God. Some understand them as rebellious angels. Others see them as powerful human rulers, or as men from Seth's godly family line marrying women from Cain's line. Whatever view someone takes, Genesis itself does not identify demons as departed Nephilim.
There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown.
Why people make the connection
The Nephilim explanation is usually drawn from ancient Jewish writings outside the Bible, particularly 1 Enoch. In that account, the unusual offspring of angels and humans die, and their spirits become harmful wandering spirits. This helps explain why the idea has endured.
Those writings can show how some ancient readers understood Genesis, but they do not carry the same authority as Scripture. Jude and Peter both refer to angels who sinned and now await judgement (Jude 1:6; 2 Peter 2:4). Some readers see an echo of Genesis 6 there. Still, neither writer says demons are the spirits of Nephilim.
And the angels which kept not their first estate, but left their own habitation, he hath reserved in everlasting chains under darkness unto the judgment of the great day.
What the Gospels say about demons
The Gospels plainly present demons, often called unclean spirits, as hostile spiritual beings under Jesus' authority. In the region of the Gadarenes, the spirits recognised Jesus and pleaded with him rather than resisting his command (Mark 5:6-10). Yet the story does not explain where those spirits came from.
Jesus also described an unclean spirit leaving a person and wandering through dry places before seeking somewhere to rest (Matthew 12:43-45). That can sound like the ancient Nephilim tradition. But Jesus was warning about a life left empty and unchanged, not giving a full history of demons.
For he said unto him, Come out of the man, thou unclean spirit. And he asked him, What is thy name? And he answered, saying, My name is Legion: for we are many. And he besought him much that he would not send them away out of the country. Now there was there nigh unto the mountains a great herd of swine feeding. And all the devils besought him, saying, Send us into the swine, that we may enter into them.
Hold firmly to what is clear
It is good to ask careful questions about passages that leave us curious. Scripture does tell us that evil spiritual powers are real, and that Christians should not treat spiritual conflict as merely human conflict (Ephesians 6:12). It also teaches that Jesus has complete authority over them.
So the honest answer is no, the Bible does not establish that demons are Nephilim spirits. You can recognise the theory as an old interpretation without treating it as certain. The clearer truth is that Christ is greater than every hostile power, and those who belong to him need not be ruled by fear (Mark 1:27).
And they were all amazed, insomuch that they questioned among themselves, saying, What thing is this? what new doctrine is this? for with authority commandeth he even the unclean spirits, and they do obey him.