I'm afraid you might have to wait much longer than what you'd like it to hear..
Personally I still think the so called "pilot shortage" is and always has been just a myth - I think there's a "good pilot contract shortage" - pay accordingly and pilots WILL come, supply and demand does work.

However, that's irrelevant here - whether there's a pilot shortage or not, no US airline that I'm aware of will sponsor a person's work visa and/or an eventual green card.

In other words, you must be a permanent resident or have some other type of work permit before they'll offer you a job. If you are a contractor and it's a temporary job things might be different but for major airlines you do need to be able to work here legally.

To Ausflyer - US airlines typically don't force their pilots to live in or near their domiciles. They'd like to but they don't. At ups we have pilots who commute to/from europe, asia and I believe we used to have someone commute to Australia too. He might've retired by now..

To summarize - work permit is a must, a degree is still preferred. Recommendations and face-to-face job fairs help tremendously.

Good luck to you all..