The school where I trained in AZ doesn´t do Visas, but maybe someone else can comment on other schools.
I can recommend Chandler Air Service, in Chandler, Arizona. I did my FAA Instrument, Commercial, Flight Instructor and Multi-engine with them. They are a part 61 and 141 school, and registered with SEVIS. I did this over the course of around one and a half years, but made a friend there from Belgium who now works in Africa. He did 0 hours through Private, Instrument, Commercial (including all the time-building) and Multi-engine in around 5-6 months.
The quality of instruction is very high at that school, and the only negative aspect of the weather is that it gets very hot in summer. The heat is bearable though, and hasn't stopped people flying throughout summer there. Airspace wise, there's Phoenix Class B nearby, with a shelf overhead, but you can stay clear of that and don't need to enter/exit it while training from Chandler. Plenty of Class G to the South. The beauty of the location allows you to avoid the major airspace when you can't be doing with it (unlike a lot of airfields over CA way) but is close by enough to be useful when you want to use it. One example would be when flying North from Chandler. You can either scoot out to the Northeast and avoid the class B by flying under it, or fly west from Chandler and then transition through the Class B to take you directly overhead Phoenix Sky Harbor.
Their website is
Chandler Air Service
@makchunyinjohnny
Most importantly FAA PPL-IR-CPL MEP takes ~ 6 months whilst the JAA (FAA PPL, IR -JAA CPL ME, ATPL ground) takes ~ more than 12 months without the conversion
Have you already done the JAA ATPL exams?? As you still need to pass them to obtain the theoretical knowledge prior to "converting" to JAA. Hope you're aware of that, and haven't overlooked such an important aspect.