Water has multiple glassy states, often called amorphous ices. Low-density (LDA) and high-density (HDA) amorphous ice are separated by a dramatic, first-order like phase transition. It has been argued that the LDA-HDA transformation connects to a first-order liquid-liquid phase transition (LLPT) above the glass transition temperature Tg. Direct experimental evidence of the LLPT is challenging to obtain, since the LLPT occurs at conditions where water rapidly crystallizes. In this work, we explore the implications of a LLPT on the pressure dependence of Tg(P) for LDA and HDA by performing computer simulations of two water models – one with a LLPT, and one without. In the absence of a LLPT, Tg(P) for all glasses nearly coincide. When there is a LLPT, different glasses exhibit dramatically different Tg(P) which are directly linked with the LLPT. Available experimental data for Tg(P) are only consistent with the scenario including a LLPT.