Depending on where you'll be in Omaha there may be closer options but I've swam a few times at The Athletic Club at 132nd and Center. Live in Lincoln so I don't swim in Omaha often.

What part of town?

Swimmers Guide is a great resource. Let us know now this turns out.
https://www.swimmersguide.com/.../index?q=omaha%2C+ne

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Karen ST Concierge

Northwest O - Common Ground (Elkhorn)
Southwest O - Lifetime Fitness
Central O - The Athletic Club as previously mentioned or JCC/Jewish Community Center
Bellevue - Lied Community Center or Offutt AFB if you have military clearance
Eastern Omaha - The Athletic Club in midtown crossing
There are of course several YMCAs throughout town as well and some of the Genesis may have pools.

If you waited until late June you could swim at the CHI Health Center for the Olympic trials ;)

Hope that helps.

Will just like the olympic trials opportunity. Oh well. Thanks for the help, folks!