In an event when two thirds of both Houses of the Congress consider it necessary, may propose an amendments to the Constitution. Further, on the application of the Legislatures of two thirds of the several States, the Congress may call a convention to propose the amendments.
There are typically two ways laid down in the US Constitution for how to propose an amendment.
The first way is for a bill to pass both the houses of the legislature, by a two-thirds majority in each. Once the bill has passed in both houses, it goes on to the States. The Congress generally puts a time limit for the bill to be approved as an amendment.
The second method is for a Constitutional Convention to be called by two-thirds of the Legislatures of the States, and that Convention proposes one or more amendments. These amendments are then sent to the states to get the necessary approval by three-fourths of the legislatures. However, in practice, this process has never ever been used.
However, any amendment must be ratified, or approved, by three-fourths of states.
It can be mentioned here that at no point the President has a role in the amendment process. He cannot veto an amendment proposal, or even ratification.
