Just so you are not at risk of misunderstanding: To travel to any country which does not have a mutual agreement with your own whereby the requirement for a passport when entering the other country is lifted - you need a passport. Where such an agreement exists, you still need to carry a recognised and accepted form of identification either on entry or whenever asked by competent authorities to prove who you are and where you are from. Examples of such agreements are the US/Canada agreement and Schengen group countries in Europe. The US/Mexico agreement (to my knowledge) works only in one direction - the Mexican authorities do not in every case require passports of US citizens but Mexicans are under considerably greater restrictions going the other way.
Where countries require a visa to be held prior to entry then the visa can only be obtained by presnting a passport for the visa to be entered into it - you may occasionally be able to get a piece of paper to say that the visa is available, but that will only be issued by referring to the passport number, etc. and the passport with the visa is required on entry.
As advised, the foreign country's embassy will be able to explain what is the actual requirement in each case.