Ok the cheapest one I've heard of is the one at the Sofitel City Centre Hotel...last I heard was 1500 for 6 months including tennis courts (or I might be mixing it up with squash courts)...
If you want a package that includes classes etc. you could try the Nautilus Academy in Musalla towers, you'd definitely get it for a lot less than what you're paying in Aviation Club...only problem is its in the heart of Bur Dubai, so I imagine the traffic would get quite choked approaching the area at peak times....but facilities are quite good and good trainers as well...as for the rates you have to negotiate, whatever they tell you at first is probably not what you'll end up paying...and if you can rope in a corporate deal, very good savings...send me a PM if you pursue this and they don't get back to you, some of my friends are halfway through the haggling process with awesome results
a slightly more expensive option is this place called ActivePlus opposite American Hospital (near Movenpick)...but I think you'd still find it a little cheaper than 4000, with state of the art facilities (they had tv screens on each exercise bike!)...Hmm if you're willing to travel as far as Knowledge Village, The Shapes Health Spa has things like classes etc as well in addition to the gym (I actually just won a 3 month membership there, I think the 3 month membership is valued at 1000 Dhs...but no swimming pool)
Some of the Deira options are cheaper as well I think, Bustan Rotana (opposite Millenium Airport Hotel) offers a really competitive deal if you work with Emirates (half price I heard?) and there's this place bang opposite Al Mulla Plaza, in the Mamzar Shopping Center called The Platinum Health Club
Am pretty sure you'll be spoilt for choice in your quest....4000 is way above the norm and there are many cheaper options