For the FAA ATP, yes. Do it on a piston single. Find a small school that is SEVIS approved, in a quiet region, that's relatively cheap and has access to an FAA examiner and has both precision & non-precision approaches nearby. The precision will be an ILS, the non-precision can be anything.
* Jump through the TSA insecurity bull!!!!.
* Do the ATP theory exam
* Whatever amount of flying training in the single engine aircraft you need to be able to pass the test.
* Do the flight test in a single with the FAA. It's free with the FAA, unlike using a Designated Pilot Examiner.
Your FAA ATP will then be valid only for single engine aircraft (land or sea, as appropriate) but it's still an ATP. I did a single engine ATP for both land and sea for the fun of it. Did the test in a C180 amphibian that was land based. Because we did the necessary water & land landings I got both licences from the one flight test.
**If you hold a foreign professional licence there isn't a requirement for an instructor recommendation. You could do the flight test without any training. I wouldn't recommend it unless you're very familiar with the US way of doing things. Remember, the US system uses TERPS, not PANS-OPS criteria.
Other points:
Get a PTS for the ATP. Study it to know what you have to do in the test. Don't forget the FAA ATP test standard is quarter scale tolerance, not half.
Expect an oral portion to your flight test where you'll be given operational scenarios at that licence level and expected to apply the rules correctly.
Also, make sure your hours meet the FAA's ATP requirements as set down in the FARs. They don't care about another jurisdictions requirements, only their own.