A tad on the pesimistic side i think the NG 737 can get ETOPS approval it has to have 2x fmc the APU that will start and run at FL410 and the big crew O2 tank.
Some seats would have to come out as i doubt that it would make it with the full 189 pax.
The NG 737 has long legs im told that a new aircraft was flown direct from boeing field to berlin this was of course without pax but it shows that it is not a short range aircraft.
If the trans atlantic thing will make money is another matter as the ETOPS set up costs are high and the pax load will be lower than normal but it is not imposable.