If you go for the A320, stupid as it sounds, you'll need two TSA checks. One for the first issue of a FAA certificate and one for the type. If you're current in light twins, figure 3 training flights and checkride for the ATP.
If you have less than 500 hours MEL turbine time your first type rating comes with a Supervised Operating Experience limitation. For the first 25 hours you have to fly with a rated Captain. When you can show 25 hours on type you can go to a FAA office and have the limitation removed. With less than 500 hours MEL turbine the type class will be about 2 weeks classroom and a week of sims.
If you have the 500 hours turbine but no time on type I know one 737 school sends you study materials and the classroom training is only a week plus the week of sims.
If you go for the 737 the FAA only issues one type rating good for -100 through -900. Many foreign CAAs issue a separate type for 737NGs. Most US 737 schools training in 300-400-500 sims or even 100-200 sims. Then if you want the NG differences it will be extra days and money. Or you can pay more and do the training in NGs from the start.