Recent studies reported the observation of prompt elastogravity signals during the 2011 M9.1 Tohoku earthquake, recorded with broadband seismometers and gravimeter between the rupture onset and the arrival of the seismic waves. Here we show that to extend the range of magnitudes over which the gravity perturbations can be observed and reduce the time needed for their detection, high-precision gravity strainmeters under development could be used, such as torsion bars, superconducting gradiometers or strainmeters based on atom interferometers. These instruments measure the differential gravitational acceleration between two seismically isolated test masses, and are designed to observe signals around 0.1 Hz. We show that these instruments should be able to detect prompt gravity perturbations induced by earthquakes larger than M7, up to 1000 km from the earthquake centroid within P-waves travel time and up to 120 km within the first 10 seconds of rupture onset, provided a sensitivity in gravity strain of 10^{-15} Hz^{-1/2} at 0.1 Hz can be achieved. The analysis involves simulations of the expected gravity strain signals based on an analytical model of gravity perturbations generated by fault rupture in a homogeneous half-space. As an immediate application, we discuss the possibility to improve current earthquake-early warning systems (EEWS). Our results suggest that, in comparison to conventional P-wave-based EEWS, a gravity-based warning system could perform faster detections of large off-shore subduction earthquakes (at least larger than M7.4). Gravity strainmeters could also perform earlier magnitude estimates, within the duration of the fault rupture, and therefore complement current tsunami warning systems.