Ba3BiIr2O9 and Ba3BiRu2O9 undergo very unusual structural transitions at 74 and 176 K respectively, with instantaneous negative thermal expansion of up to 1% in the Ir compound. These transitions are closely associated with the opening of gaps in the Ir/Ru spin excitation spectra. Here, we to explore the ability of external applied pressure to suppress the transitions. The response of the structures to pressure will shed light on the driving force - i.e., whether structural modifications drive the system to change spin state; whether quantum instabilities in the spin state drive the structural transition; or whether some other local factor (possibly charge transfer or charge disproportionation) drive both macroscopic effects. The value of the pressure required to suppress the transitions to precisely 0 K will also provide an independent measure of the energy of the spin gap opening.