With current theory: no. (There isn't even a way to make "ordinary" exotic matter in current theory!) Current theory makes it look very likely that all exotic space-time configurations decay very quickly to more boring configurations.
So it all depends on how fast you're willing to wave your hands. The no-hand-waving-at-all answer is that current physics-as-we-understand-it requires exotic matter or something equally astonishingly massive moving at high speeds to produce major warpage of space.
Wave your hands at moderate speed, and you could claim that black holes don't evaporate -- it's a theoretical prediction that has never been observed either directly or indirectly -- and that something like Robert Forward's black hole fluid (from Dragon's Egg) can be made and used. (I don't think this can yield an Alcubierre warp drive directly, but wave your hands a bit faster and explain how they can be used to create blah blah's which combined with left-parity whatsits make it possible produce a warp drive without stable exotic matter...)
Wave your hands still faster and say that you've found a source -- perhaps made by a long-transcended alien civilization -- of nanoscopic space-time knots which are stable, being protected from decaying by their topology. And stable nanoscopic space-time knots can do just about anything....
Finally, there's the long-sought merger of General Relativity with quantum theory. Whatever form it takes, it's going to provide some new interactions which can affect gravity. Since we really don't know much about the unified theory, you can get away with a lot there still. (Especially emergent space-time theories -- it's actually kind of plausible that if space-time itself is an emergent phenomenon rising out of something deeper and quite different, that some new technologies might arise which allows space-time manipulation we can't presently foresee.)