Though I've not heard of doing it, if you currently have a 308, I suspect your gunsmith recommended a 375/08 which could be built on most rifles in 308. The '06 case is longer and probably won't function through the shorter actions used for most 308s.
True Whelen's are usually found in 35 cal, and rarely a 40 cal. The problem that occurs with the larger calibers in the narrower cases like the '06 and '08, is that larger diameter bullet leaves very little shoulder on the case in which to head space the cartridge to in the chamber. You end up with almost a straight walled case that can't build up enough case pressure to keep the velocity up. A sharper shoulder like an Ackley Improvement, would help with the cartridge sealing at the shoulder.
The 35 Whelen is a great Elk gun.