I'm not sure if I understand your question correctly, you need a visa to remain in the United States, if it's for a year or 10 years whatever the occasion is there are special visas.
A greencard isn't really a visa, greencard holders basically have the same rights to work as a US citizen. However when you want to enroll in flight training be it 61 or 121 etc you have to get a background check done, that is accomplished by TSA and the company you work for. TSA relays your info to you and your company and all your company has to do is approve you for work, and state that they are indeed hiring you to fly for them.
So to kind of answer your question (or at least I hope to have) regardless of what country your from you need a visa.