We present an all-sky 90% confidence level upper limit on the cosmic flux of relativistic magnetic monopoles using 2886 days of IceCube data. The analysis was optimized for monopole speeds between 0.750c and 0.995c, without any explicit restriction on the monopole mass. We constrain the flux of relativistic cosmic magnetic monopoles to a level below 2.0×10^{-19} cm^{-2} s^{-1} sr^{-1} over the majority of the targeted speed range. This result constitutes the most strict upper limit to date for magnetic monopoles with β≳0.8 and up to β∼0.995 and fills the gap between existing limits on the cosmic flux of nonrelativistic and ultrarelativistic magnetic monopoles.