What happens before congress can process an amendment to the constitution?

It's not clear what you mean by process. Congress has no veto power over an amendment once 3/4 of the states ratify it. Neither does the president nor the supreme court. Congress never votes on or ratifies an amendment in the process of making it law. Before an amendment goes to the states for consideration, the Senate tells the states whether they have to ratify via the state legislatures, or by way of state conventions.

Amendments can originate in Congress, but they can also originate in the states themselves, without input from Congress other than the ratification process that the states must use.

The 'States' origin of an amendment has never been used, and some say, perhaps with reason, that congress has stalled many legitimate attempts by the states to get the amendment process going. That would be unfortunate, and in my opinion, reason for the states to put serious legal pressure on congress. After all, congress would be motivated not to allow the states to mess with the system that keeps them all comfortably employed.