Anecdotal evidence now exists for rapid, large amplitude, X-ray variability in a few AGNs, all having unusually narrow Balmer lines. In order to explore in a systematic manner the possible relation between line width and X-ray variability properties we initiated a pilot program using the ROSAT HRI, where we monitored the X-ray luminosity of a small sample of six quasars on timescales of 1 day to 6 months. The six quasars were extracted from a complete sample of optically selected quasars and lie at the two extreme ends of the PSPC spectral index distributions, and hence of the H$\beta$ FWHM. These observations allow us to test the suggested dependence of variability on X-ray slope. Here we report on the results for the 1 day to 1 month timescales. All three steep PSPC spectrum quasars show significant variability, with a typical amplitude of $\sim 20$\% on 1 day timescale, rising to a factor of $\sim 2$ on 20 days timescale. All three flat spectrum quasars show typical variability of 20\%, or less, on all timescales. These results suggest a remarkably tight relation between the X-ray variability properties of AGNs and the X-ray spectral slope. Such a relation agrees with the hypothesis that steep spectrum, narrow line, quasars are emitting close to Eddington Luminosity, making them the analogs of Galactic BHC in the high (and soft) state.