From what I understand/have been told from the new TSA rules
You need a student visa (MJ1 or something like that)
For this, the school needs a curiculum, which mean part 141.
Then you need to submit $130 and some info so the TSA can do a background check.
If the airplane is under 12.5, you can start your training right away, but must stop if the TSA doesn't clear you. If it's over 12.5, then you need to wait for approval from the TSA to start your trainning
Okay, I'll plead ignorant on what is required for getting a student visa in this country, but from what I've read on the above posted link, you can have a background check done (required) and start training. Now, I don't know what's required to STAY in this country, but the question was can you train here and yes, you can train here under part 61 or 141.
And, all these questions show why the TSA, FAA, and any other acronym should be doing these checks, not us CFI's. I'm hoping AOPA can get some of this straightened out with them at alphabet soup.
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So if you have the green card you still need to do this? It was said above that a student visa is needed. But what's the use if I already have the green card?
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I also plead ignorant to the green card issue. I wish I could help you out, but I am unfamiliar with such issues. However I am postive that once you get your TSA stuff submitted you can train under 61 or 141. (as long as no red flags pop up) Our school already has some folks rolling under both programs.
I do seem to remember that being 141 does help to get a visa to get into the USA?! Any know about this? The folks that are training with us so far seem to be Permanent Residents already. I don't believe they are on temporary visas. But I could be wrong. I'll try to ask around for ya...