Yep, not a problem. I had my Rem 700 30-06 rechambered to 7mm mag. Opening up the bolt face is not a problem, you will probably want to fit a sako style extractor while you are at it.
I agree that you can easily do it but there are some other things to consider. You will have to do some feed rail work and you probably have a 22" barrel on your '06 which pretty much defeats having a magnum in my opinion. You could put a 26" barrel on your '06 and probably get as much or more velocity than you would with a 22" 300. Not suggesting that you do that, just trying to make a point. Good luck with what you decide.....Rich