FWIW, "having a barrel reamed" usually means reboring the entire barrel to a larger internal diameter and then rerifling it.
A .338 Win Mag barrel is already .338", so it could only be reamed out to a larger size - the next larger which is .358" (.35 caliber).
If you meant reCHAMBERED, then a barrel's chamber area can only be made larger - in fact large enough to erase all trace of the previous chambering - and sometime bolt face changes get involved also.
Since the .338-06 is based on the .30-06 case necked up to .338", and the .338 Win Mag is a much fatter/longer belted magnum, is you want to make a .338-06 out of a .338 Win Mag, the rifle must either be rebarreled or the barrel set back, rethreaded, rechambered, and the bolt changed to a smaller -06 size.
If all you have is a bare .338 Win Mag barrel, then the entire chamber end must be cut off, shortening the barrel, the end rethreaded for whatever action/application, and then chambered for .338-06.
The resultant reduced barrel diameter in the area of the new chamber reinforce may not be suffucient for safety - if it will even be safely large enough for the action threads. YMMV
The resultant cartridge feeding issues on an installed barrel, due to internal rifle/mag difference for a belted mag case and a smaller non-belted case, would also need work.
The bottom line, since it's not economically viable, the answer to your question is "no".
The other way around, though - .338-06 to .338 Win Mag - is a little more "doable", still needing action/bolt work at the least besides the rechambering.
.
