What Scripture calls angels
The clearest biblical description of angels is that they are spirits sent by God to serve his purposes. They are not merely forces or symbols. Angels speak, obey, worship, and carry out God's commands, which shows that they are personal beings (Hebrews 1:14; Psalm 103:20).
Yet the Bible's usual word for them is not soul. That matters because Scripture uses its words carefully, even when everyday English uses soul more broadly to mean the inner life of a person.
Are they not all ministering spirits, sent forth to minister for them who shall be heirs of salvation?
How the Bible speaks of human souls
In Genesis, God forms Adam from the dust, gives him the breath of life, and Adam becomes a living soul (Genesis 2:7). The Hebrew word behind soul is nephesh. It can refer to a living creature or life itself, rather than simply an invisible part tucked inside a body.
Scripture also speaks of the human spirit returning to God at death (Ecclesiastes 12:7). Christians have sometimes used soul and spirit in slightly different ways, while others use them more interchangeably. Either way, the Bible gives fuller teaching about human souls and spirits than it does about the inner makeup of angels.
And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.
Angels are real persons, not human souls
Angels can understand and respond. They long to look into God's saving work (1 Peter 1:12), and there is joy among them when a sinner repents (Luke 15:10). Those details show awareness and response, even though the Bible does not label that inner life a soul.
They are also distinct from human beings. Jesus says that those raised from the dead do not die any more and are like the angels in that respect (Luke 20:36). People do not become angels when they die. God has made different kinds of creatures, and angels belong to his heavenly host (Psalm 148:2; Psalm 148:5).
A careful answer
So, do angels have souls? If soul means a personal inner life, someone might use the word that way in ordinary conversation. But if the question is whether the Bible specifically teaches that angels have souls as humans do, the honest answer is no. Scripture calls them spirits and does not explain their nature beyond what we need to know.
That limit is not a problem. The Bible directs our attention less towards mapping the unseen world and more towards the God whom angels worship. They are servants of God, not beings for us to fear, seek, or worship (Revelation 22:8-9).
And I John saw these things, and heard them. And when I had heard and seen, I fell down to worship before the feet of the angel which shewed me these things. Then saith he unto me, See thou do it not: for I am thy fellowservant, and of thy brethren the prophets, and of them which keep the sayings of this book: worship God.