The possible origin of millisecond bursts from the giant elliptical galaxy M87 has been scrutinized since the earliest searches for extragalactic fast radio transients undertaken in the late 1970s. Motivated by rapid technological advancements in recent years, we conducted $rm simeq 10~hours$ of L-band ($rm 1.15-1.75~GHz$) observations of the core of M87 with the Arecibo radio telescope in 2019. Adopting a matched filtering approach, we searched our data for single pulses using trial dispersion measures up to $rm 5500~pc~cm^{-3}$ and burst durations between $rm 0.3-123~ms$. We find no evidence of astrophysical bursts in our data above a 7$sigma$ detection threshold. Our observations thus constrain the burst rate from M87 to $rm lesssim 0.1~bursts~hr^{-1}$ above $rm 1.4~Jy~ms$, the most stringent upper limit obtained to date. Our non-detection of radio bursts is consistent with expectations of giant pulse emission from a Crab-like young neutron star population in M87. However, the dense, strongly magnetized interstellar medium surrounding the central $sim 10^9 M_{odot}$ supermassive black hole of M87 may potentially harbor magnetars that can emit detectable radio bursts during their flaring states.
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