I've never heard of any club authorized to issue visas, because they are clubs, not flight schools. Most flying clubs require you to be a member, then they rent you the plane. Then you hire your own instructor who usually also has to be a member. An easy thing to do, is contact a Flight Standards District Office in the area you want to train, and ask them who is authorized to accept foreign students.
You can find these offices on the FAA.org
Also, going into flight training with the idea of trying to save money is like trying to buy the cheapest Ferrari you can find, it may be cheaper, but it's till expensive. A good school will save you money in the long run, however the key is the instructor. In the US, pilots need 1500 hrs to get into the airlines, so it should be getting a bit easier to find higher time instructors, in the past, at 500 hrs, they were gone, sometimes even sooner. Using your PC to introduce yourself to flying. It is a really good tool, when you get into instrument flying.