It doesn't sound like you are from the US, so it's not quite clear what you mean by "off-piste." In the US, off-piste means basically ungroomed runs inside the ski area boundaries. Because of that, you have all of the ammenities of a ski area, including hotels, restaurants (with their inflated prices), etc. Most importantly, it also means that ski patrol will provide rescue and provides avi control, marks hazzards (like cliffs), and closes areas that are considered unskiable or dangerous (with some exceptions). It's quite a bit different from what I understand to be off-piste in Europe.
If you mean that you want to go backcountry (outside of the ski area boundaries), then that's another matter.
Resort off-piste skiing close to Denver is quite different from area to area. Arapahoe Basin and Loveland are two areas that have a lot of off-piste skiing, but neither has any lodging at the resort, so it can be a bit more of a challege to get to the area than the other resorts if you do not wish to hire a rental car. Vail's off-piste skiing has a tremendous amount of area, but it mostly is flat compared to other areas. Breckenridge has a lot of high alpine steep terrain and a fair amount of tree skiing. Beaver Creek has a lot of long steep bump runs, some good tree skiing, and small amount of more extreme skiing. Copper has "naturally divided terrain" where the expert off-piste skiing is served by separate lifts, meaning that there usually is no lift line for much of the interesting terrain. Keystone, in my experience, has much less in the way of off-piste terrain. The Aspen resorts have fantastic off-piste skiing; particularly Highlands where you can hike Highlands Bowl for some of the signature terrain in Colorado or ski amazing lines without a hike. Snowmass and Aspen Mountain have great off-piste terrain as well. Any of these resorts can be accessed by van service, although it is a bit pricey.
If you mean backcountry, then there's many many choices, but you will need a guide to assist you. It requires ski touring; not sure that's what you want. I know little about ski touring in general or about guides in Colorado specifically.
If it was me, I'd pick Aspen. I think it has some of the best expert skiing in Colorado. It's far removed from the resorts close to Denver, meaning that it will be much less crowded than the other resorts I've mentioned. With 4 mountains and easy transportation between them, you've got more terrain than you can explore in the time you'll be here. And the food is great, there's lots of other stuff to do, and its downright beautiful as well. There's not too many better views than from the top of Highlands, or even from the Cloud 9 restaurant at Highlands.
There's other choices as well, but hopefully this will give you a start. If you'd like a bit more help, then it'd be good for you to describe what type of skier you are, what type of terrain you are interested in, and what resort ammenities you're looking for.