Any private onsens that permit suits?

I would like to spend time in an onsen with a private group of mix gender friends, but we have no interest in bathing together nude. Can we relax in a private onsen and still wear bathing suits?

What you do behind closed doors in a private onsen is your business, but keep in mind that you are supposed to keep towels and suits out of the bath water for various reasons. One being that the fabric can react with the bath water and degrade the quality of the bath.

I understand how you feel somehow but... as a student I went to Finland for a whole summer for a practicum in civil engineering. The very first day at work, right after lunch, I was invited to take a sauna with the 2 dozen guys I would work with. The sauna was very hot and my towel was yanked off my lap "for health reasons". After 15 minutes we all went out and walked down the beach to jump in the sea. That was a bit unnerving, especially for a very shy geeky 21 years old guy, but after doing that daily i soon enjoyed it. 2 weeks later one of the workers invited me home for the weekend. His home was by the sea and a soon as we arrived we took a sauna with the other men in his family. we went out for the compulsory dip in the sea BUT THIS TIME the rest of his whole extended family.. grandma, wife, sisters in laws, nieces etc. were on the beach, all without clothes.
I froze in sheer panic then, to HONOUR MY HOME COUNTRY by showing we weren't chicken I slowly walked past them to the sea and back...
AS TOURISTS we have the duty to respect the customs and regulations of the country we visit. If onsens don't allow bathing suit then we cannot go there. It is that simple!

You could make a compromise and go to an onsen town with a foot bath...or a ryokan that has one. Taiseikan in Hakone, for example, has the usual separate-gender baths inside & out, but then has a nice footbath out in a wooded setting. I went with a group of friends and after relaxing separately in the baths, we went outside in yukata and sat outside with our feet soaking...not quite the same, but a nice experience.

Thank you for your imput everyone, this has quite helpful.

It's not private but have you tried onsen like this? I think it would be more comfortable this way for your ..maybe.
I like this one, there's a pool you can swim in year around. Nothing better than hot spring water pool.

http://www.kankou-gifu.jp/en/hotsprings/takayama.html