Only the Congress (which is in the legislative branch) can propose a constitutional amendment, which must not only be approved by a large majority in the Congress itself but must also be ratified, or approved, by two-thirds of the states.
Since one of the powers of the president (who is in the executive branch) is to suggest legislative agendas, the president can technically also suggest a constitutional amendment to the Congress. But in this case the president's proposal is merely a suggestion and has no legislative authority, since the real authority is with the Congress.
Answer: The legislative branch