go and measure the twist of the rifle you are considering buying. Then you will know for sure which one it is.
Remington introduced the .244 with a 1-12 twist. Great for varmint shooting. Unfortunately, the market wanted dual purpose 6mm caliber rifles, and the .244's twist usually didn't do well with the regular 105gr "deer" bullet.
Sales never really took off, and lagged badly behind the .243Win. After a couple of years, Remington changed the twist on its .244s to 1-10". These rifles will shoot acceptably well with the 105gr bullets.
A few years later, Remington tried to give the .244 a new lease on life, by changing its name to the 6mm Remington. 6mm Rem rifles have a 1-9" twist.
Just checked, a Rem 722 will have a 1-12" twist in .244 Rem. The faster twist .244s seem to only be in the model 700.
As many others have mentioned, the only way to be certain your (prospective) rifle will not shoot acceptably well with the 105gr bullets is to shoot it. It is quite possible that it will shoot acceptably well with "deer" bullets, but not as tight grouping as with lighter varmint class slugs.
If the rifle is suitable in all other aspects, I'd say get it. Then try several different loads (brands/styles of bullet and different powder combinations) and see how it actually performs. It may be a 3/4 MOA rifle with an 80gr bullet, and only 2MOA with a deer load. That might be good enough for your needs. Or it might not.
The 6mm bore is light for deer, although with careful choice of shots and proper bullet construction (for what you are shooting through) it has, and will do the job on game as large as elk (neck/head shots are best).
Raking shots on whitetail & muleys are iffy, and ought to be avoided. Now, there are tons of folks who have dropped their deer with these shots, and they may write in to argue. But there have been many failures as well, and the 100gr-ish deer load in the 6mm bore size doesn't have a lot of extra, so one shouldn't expect it to do what it was never meant to do.