According to 1 Samuel 9-10, Samuel the prophet anointed Saul as the first king of Israel. God directed Samuel to anoint Saul, a tall young man from the tribe of Benjamin, after the Israelites demanded a king so they could be "like all the other nations" (1 Samuel 8:5). Samuel privately poured oil on Saul's head and declared that God had anointed him ruler over His people Israel (1 Samuel 10:1). Saul was later publicly presented to the people at Mizpah.
The other prophets named in the options served in different periods. Nathan was a prophet during the reign of King David, not before the monarchy began. He is known for confronting David over the sin with Bathsheba (2 Samuel 12). Elijah was a prophet who served during the divided monarchy, primarily opposing King Ahab and Queen Jezebel in the northern kingdom of Israel (1 Kings 17-19). Elisha was Elijah's successor and continued prophetic ministry in the northern kingdom (2 Kings 2-13). None of these three anointed Saul.