Agence Cholleton outside Paris makes bespoke rugs - anything your heart desires if your wallet can support it. Wilton, axminster, handwoven, etc. There are several Wiltons in white or ivory on their website. Fibers are wool, wool and silk, and linen.

Here is a link that might be useful: Agence Cholleton web site

Home Decorators seems to have several alternatives for creating your own. The one I have linked would be a more traditional feel, with the three rectangular fields.

custom rug

You don't say what size you need, what you want to spend, or what you mean by 'texture for interest". Do you want a sculpted pattern in the rug, or do you want the rug fibers to have texture like a friese? I assume from your description that this is going in a traditional setting - is there a reason you want an all white area rug, versus something with some color?

How about this one by Martha Stewart?

Here is a link that might be useful: Martha Stewart rug

I love sheepskin rugs. I think they work well in many settings - primitive, modern, country, or even, when used as accents, simple traditional. To me, they say something completely different from shag rugs. Costco has 'em, in sizes from quad on up to at least octo, maybe more.

If you need typical traditional, of course, they won't work.

You could also buy white broadloom, have it cut to size and bound. I have done this a number of times, and it usually works out to cost less than a comparable quality area rug.

palimpsest beat me to it -- in our case, we've gotten *wool* berber remnants cut and edged to make very nice, clean-lined area rugs at less cost.

I am echoing palimpsest and flyleft. Just had an off-white Karastan wool berber cut to measure. I did have it bound with a wide mitred cotton canvas border. The border cost almost as much as the rug, but for $900 I got a 8'9" x 11'9" good quality wool rug made just the way I wanted - and it included the rug pad. I bought it at a FloorsUSA.

I find that although it may be convenient to order on-line, I really like to see and feel products in person!

Good-luck in your search!