A residency program cannot sponsor permanent residency (green card status). There are 2 major types of Visas used by physicians studying in the U.S., the J-1 Visa which allows you to study in an U.S. residency or fellowship program but then requires you to return to your home country for two years before being admitted to the U.S., or an H-1B Visa where you are working as a technical specialist.
H-1B Visas are very difficult to get for residency programs. Usually only Canadians manage to get them because of the similarity of medical education and licensing in the U.S. and Canada.
J-1 Visas are much easier to get. The tricky part is that J-1 Visas can't be converted into other types of Visas under ordinary circumstances, so you need to be prepared for the 2 year home residency requirement. However, every year there are a limited number of waivers granted for the 2 year home residency requirement.
Waivers are not easy to get. You must have a job offer from a facility that is elligible to sponsor a waiver, (generally a Health Provider Shortage Area)and then you need to submit an application which must be approved first by the state department of health, and then the U.S. department of State, and finally the I.N.S. And then, even after your waiver is approved, you will still need to get an H-1B waiver.
The company I work for has a team of specialists that assists with the entire J-1 waiver process - from matching you with an eligible employer, preparing your application, and tracking it through the administrative maze to approval.
Right now, there isn't a lot of information on this on our website, but the newly designed site is scheduled to launch on Friday. You might want to check it out some time after that. We are at
<a href="http://www.practice.org" target="_blank">www.practice.org</a>
As far as qualifying for a U.S. residency program and the J-1, there is a lot of information at
<a href="http://www.ecfmg.org" target="_blank">www.ecfmg.org</a> That's really the place to start.
Good luck in your process!