I admit to knowing nothing on this matter and ask my question to those of you that do.
I own a Remington 700 Classic chambered for 338 WM is it possible to have the barrel changed to a 375 H&H or to have it rechambered to 375 H&H and an approx. cost?

Absolutely! Replace the barrel, fit it in the stock and you're done. Cost would vary based on brand, type of barrel and any extra's.

Remington actions are very long and all you need to do is change the barrel to a .375 H&H chambered barrel. It must be professionally fitted and blued and you should be ready to shoot.

Your .338 barrel may be rebored but a new barrel might be the best bet.

I thank you both for the replies. I'm hesitant to do the switch though as I really love this rifle just as it is. I may just look to purchase a used 375 H&H though now I know I have an option should I wish to go that route. Thanks again!