Mount Horeb is an alternate name for Mount Sinai. Though there is a little debate about this, the majority of scholars consider them to be the same mountain due to the general interchangeability of the terms throughout Scripture.
“In many instances “Sinai” and “Horeb” are used synonymously; where a distinction appears, the mountain itself is Sinai and the neighboring wilderness area bears the wider designation Horeb…In Ex. 3:1, where the term Horeb first appears, Moses is said to come to “the mountain of God ḥōrēḇāh.” The directional h at the end of the word for Horeb suggests that Horeb was not the name of the mountain but of the area in which the mountain was located. This leads to the suggestion that both Horeb and Sinai are names for the general area, Horeb clearly a Semitic term and Sinai possibly a name used by some other people.” (Bromiley, Geoffrey W., ed. The International Standard Bible Encyclopedia, Revised. Wm. B. Eerdmans, 1979–1988.)
The word “horeb,” חֹרֵב, occurs 17 times in the Old Testament: Exodus 3:1; 17:6; 33:6; Deut. 1:2, 6, 19; 4:10, 15; 5:2; 9:8; 18:16; 29:1; 1 Ki 8:9; 19:8; 2 Chron 5:10; Psalm 106:19; Mal. 4:4.
The word “sinai,” סִינַי, occurs 35 times: Exodus 16:1; 19:1; 19:2, 11, 18, 20, 23; 24:16; 31:18; 34:2, 4; 34:29, 32; Lev. 7:38; 25:1; Lev. 26:46; 27:34; Num. 1:1,19; 3:1, 4, 14; 9:1, 5; 10:12; 26:64; 28:6; 33:15, 16; 33:2; Judges 5:5; Neh. 9:13; Psalm 68:8; 68:17
Let’s take a look at some of the verses. Horeb is called “the mountain of God,” (Ex. 3:1; 1 Kings 19:8). It is where God made a covenant with his people at Horeb (Deut. 5:2; 2 Chron. 5:10) which is reflected in God giving the two tablets of the testimony to Moses on Mount Sinai (Ex. 31:18). Horeb is where the Israelites made the golden calf (Psalm 106:19) and we know that they did this at Mount Sinai (Ex.32:8; 34:2).
Though nothing in the Bible says that Mount Horeb is another name for Mount Sinai, the biblical evidence strongly suggests that they are one and the same.
