Does your $10K include the stone and setting?
What shade of blue does she like? A bit of violet in the blue, pure blue or with some green? Darker or lighter blue?
What shade of green does she like? A bit of yellow in the green, pure green, or with some blue? Darker or lighter green?
100% untreated or heat is okay for the sapphire?
100% untreated for the emerald, oil or resin? What level of enhancement is acceptable?
What shape and mm size are you considering?
Okay, $10K just for an untreated darker pure blue sapphire or untreated pure green emerald. Just as an FYI, top colour for sapphire is slightly violetish blue and for emerald is slightly bluish green. It sounds like you prefer a round cut only? Given the specifications, your best bet is a blue sapphire because it is easier to find and less expensive than an emerald.