N1, there are two ways to go about it. Either way will require you to get a validation letter from your CAA and pass a TSA background check. Then you could come to a training center in the US, give them 7-8 thousand dollars and in about two weeks you should get an ATP with B-737 on it. Keep in mind that in FAA land there is only one B-737 rating, good for the -100 through the -900. Most of the training centers use Jurassic or Classic 737 sims. I'm guessing if you wanted to complete the training in an NG sim you would be looking at $10K. Training centers in the US see a lot of foreign students and know all the paperwork details.
Or, with the validation letter and background check, you can receive a 61.75 "based on" private certificate with your type rating limited to VFR only. If you pass the FAA instrument written they will authorize instrument privileges on your 61.75 certificate. And IF, IF you can show that the person that trained you for the type rating on your foreign license was an authorized FAA instructor, the FAA will remove the VFR only limitation from your private B-737 rating. Then all you have to do is pass a regular ATP checkride in a PA-34 or CE-310 or some twin and the B-737 rating will transfer over to your unrestricted ATP.