The first mention of a soul in the Bible is in the context of the creation of the first man, Adam, in Genesis 2:7: “Then the LORD God formed a man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being” (or “a living soul” in the KJV). Unlike what He had done with His previous creations, God made people in His image (Genesis 1:26–27) with His own breath within them. This non-physical part of a person is usually referred to as a soul (Hebrew nephesh).
There are passages that indicate that God does have a soul: Leviticus 26:11 and Judges 10:16 use a form of the word nephesh in relation to God. And in Jeremiah 32:41, God makes a promise concerning Israel: “I will rejoice in doing them good and will assuredly plant them in this land with all my heart and soul.” However, Scripture also refers to God having a hand or a face, applying human qualities to God in a figure of speech known as anthropomorphism. It could be that biblical descriptions of God’s “soul” are anthropomorphisms similar to descriptions of God’s “hands.” So, we must be careful about saying that God has a soul. God is Spirit (John 4:24), but nowhere in Scripture is it said that God is Soul or that He literally possesses a soul.
When we consider God the Son, we can be more certain. Jesus was (and is) fully God and fully man. When the Son of God became incarnate, He took on a sinless human nature, and this included a truly human soul. In His agony in the Garden of Gethsemane, Jesus said, “My soul is overwhelmed with sorrow to the point of death” (Matthew 26:38). His human nature—including His soul—recoiled at the thought of crucifixion and all it entailed.
Much of this discussion depends upon how one defines the word soul. If we equate the word soul with personhood, then, yes, God has a soul; He is a person in that He is a being who possesses a mind, emotion, and will. If we view the word soul as the ability to express emotions, then, yes, God has a soul—He is not “soulless” in the sense of having no feeling. But we normally use the word soul in the context of humanity. In fact, some would define the soul as that immaterial part of us that links the spirit with the body. The Father is not human. He is spirit; the Holy Spirit is also immaterial; the Son has a human body and a human soul/spirit, because He is a true a human being, the God-Man who makes intercession for us (Hebrews 7:25).