Most training establishments (at least in my experience 737CL) stipulated an ICAO CPL/IR as minimum, no ME requirement I think.
Perhaps even just a PPL(A) in some places (not sure if it has been done though, SE/ASEL PPL/IR, yes of course... PPL.... think it has been done and in Europe as well, a Dutch TV program, presenter held just a PPL and was doing a 737NG TR and touch and go's on a JetAirfly 737-800 with fellow presenters filmed terrified in the back*).
*As shown on Dutch TV a few years back.