Horeb (sometimes stated as Mount Horeb) was only stated in the Old Testament where several events occurred during the Exodus. No specific location was given; but the passages that mention it suggested it existed somewhere on the Sinai peninsula. Because the same events were mentioned as those that happened during the Exodus, Horeb was probably a name for the region where Mount Sinai was.
Both Exodus 3:1 and 1 Kings 19:8 refer to it as "the mountain of God".
Exodus 17:1-7 gave the location not far from Rephidim where Moses struck the rock to get water as directed by God.
1. The whole Israelite community set out from the Desert of Sin, traveling from place to place as the LORD commanded. They camped at Rephidim, but there was no water for the people to drink.
2. So they quarreled with Moses and said, "Give us water to drink." Moses replied, "Why do you quarrel with me? Why do you put the LORD to the test?"
3. But the people were thirsty for water there, and they grumbled against Moses. They said, "Why did you bring us up out of Egypt to make us and our children and livestock die of thirst?"
4. Then Moses cried out to the LORD, "What am I to do with these people? They are almost ready to stone me."
5. The LORD answered Moses, "Walk on ahead of the people. Take with you some of the elders of Israel and take in your hand the staff with which you struck the Nile, and go.
6. I will stand there before you by the rock at Horeb. Strike the rock, and water will come out of it for the people to drink." So Moses did this in the sight of the elders of Israel.
7. And he called the place Massah and Meribah because the Israelites quarreled and because they tested the LORD saying, "Is the LORD among us or not?"
While Deuteronomy 4:10 showed Moses and the Israelites embracing God's ways, some passages tried to bring them back into correction when they strayed as in
Deuteronomy 9:8 and 18:16.
Deuteronomy 4:15 also made reference to the burning bush.
Deuteronomy 5:2 and 29:1 spoke of the covenant Moses established between God and the Israelites.
2. The LORD our God made a covenant with us at Horeb.
Exodus 33:1-6 and Psalm 106:19 examplified Israel straying when they made the graven image of the Golden Calf at mount Horeb.
Deuteronomy 1:1-8,19:
1. These are the words Moses spoke to all Israel in the desert east of the Jordan--that is, in the Arabah--opposite Suph, between Paran and Tophel, Laban, Hazeroth and Dizahab.
2. (It takes eleven days to go from Horeb to Kadesh Barnea by the Mount Seir road.)
3. In the fortieth year, on the first day of the eleventh month, Moses proclaimed to the Israelites all that the LORD had commanded him concerning them.
4. This was after he had defeated Sihon king of the Amorites, who reigned in Heshbon, and at Edrei had defeated Og king of Bashan, who reigned in Ashtaroth.
5. East of the Jordan in the territory of Moab, Moses began to expound this law, saying:
6. The LORD our God said to us at Horeb, "You have stayed long enough at this mountain.
7. Break camp and advance into the hill country of the Amorites; go to all the neighboring peoples in the Arabah, in the mountains, in the western foothills, in the Negev and along the coast, to the land of the Canaanites and to Lebanon, as far as the great river, the Euphrates.
8. See, I have given you this land. Go in and take possession of the land that the LORD swore he would give to your fathers--to Abraham, Isaac and Jacob--and to their descendants after them."
...
19. Then, as the LORD our God commanded us, we set out from Horeb and went toward the hill country of the Amorites through all that vast and dreadful desert that you have seen, and so we reached Kadesh Barnea.
1 Kings 8:9 and 2 Chronicles 5:10 referred to the Ten Commandments in the
Ark of the Covenant:
9. There was nothing in the ark except the two stone tablets that Moses had placed in it at Horeb, where the LORD made a covenant with the Israelites after they came out of Egypt.
Malachi 4:4:
4. "Remember the law of my servant Moses, the decrees and laws I gave him at Horeb for all Israel.
The only reference to Horeb that doesn't involve Moses was when Elijah fled for his life in 1 Kings 19:8:
8. So he got up and ate and drank. Strengthened by that food, he traveled forty days and forty nights until he reached Horeb, the mountain of God.
